To find the derivative of y = cos(sin(14x-13)), we will use the chain rule.
Let's start by defining two functions:
u = sin(14x-13)
v = cos(u)
We can now apply the chain rule:
dy/dx = dv/du * du/dx
First, let's find dv/du:
dv/du = -sin(u)
Next, let's find du/dx:
du/dx = 14*cos(14x-13)
Now we can put it all together:
dy/dx = dv/du * du/dx
dy/dx = -sin(u) * 14*cos(14x-13)
But we still need to substitute u = sin(14x-13) back in:
dy/dx = -sin(sin(14x-13)) * 14*cos(14x-13)
So the derivative of y = cos(sin(14x-13)) is:
dy/dx = -14*sin(sin(14x-13)) * cos(14x-13)
To find the derivative of the function y = cos(sin(14x - 13)), we can use the chain rule. The chain rule states that the derivative of a composite function is the derivative of the outer function times the derivative of the inner function.
Let u = sin(14x - 13), so y = cos(u). Now we find the derivatives:
1. dy/du = -sin(u)
2. du/dx = 14cos(14x - 13)
Now, using the chain rule, we get:
dy/dx = dy/du × du/dx
dy/dx = -sin(u) × 14cos(14x - 13)
Since u = sin(14x - 13), we can substitute back in:
dy/dx = -sin(sin(14x - 13)) × 14cos(14x - 13)
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A cube has a volume of 27 cm3. A smaller cube has a side length of that is x cm less than side length of the larger cube. Consider the function f(x) = (3 − x)3. What does f(0. 5) represent?
f(0.5) represents the volume of the smaller box when its' side length is 0.5 cm less than the larger cube
How to find the Volume of a Cube?The formula to find the Volume of a cube is:
V = x³
where:
x is the side length of the cube
The cube has a volume of 27 cm³. Thus:
Side length of cube = ∛27 = 3 cm
Since the side length of the smaller cube is x cm less than side length of the larger cube, then we can say that the function to find the volume of the smaller cube is:
V_small: f(x) = (3 - x)³
Thus, f(0.5) is the volume of the smaller box when its' side length is 0.5 cm less than the larger cube
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Find the sum of the convergent
∑ 24/n(n+2)
n = 1
the sum of the convergent series is 8.
To find the sum of the convergent series ∑(24/n(n+2)) where n starts at 1, we can re-write the given expression as a partial fraction decomposition:
24/n(n+2) = A/n + B/(n+2)
Solving for A and B, we find that A = 12 and B = -12. So the expression becomes:
12/n - 12/(n+2)
Now, we can compute the sum for the given series:
∑[12/n - 12/(n+2)] from n = 1 to infinity
As this is a telescoping series, most of the terms will cancel out. We are left with:
12/1 - 12/3 + 12/2 - 12/4 + ... + 12/∞ - 12/(∞+2)
The sum converges to:
12 - 12/3 = 12 * (1 - 1/3) = 12 * 2/3 = 8
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1. The following data show weight (in kg) of 24 women in a study: 46. 4, 53. 2, 52. 8, 42. 0, 50. 8,
43. 0, 51. 9, 59. 2, 55. 1, 38. 9, 49. 7, 49. 9, 43. 1,42. 2, 52. 7. 49. 8. 50. 7, 44. 8. 49. 2, 47. 7, 42. 9,
52. 9, 54. 1, 45. 4.
Prepare the following:
I.
Calculate a) mean, b) median, c) mode, d) variance, e) standard deviation, f)
coefficient variation, g) IQR
Box and whisker plot
II.
III.
Discuss the distribution of these data
The mean is 48.47 kg, median is 49.55 kg, mode is not available, variance is 34.1 kg², standard deviation is 5.84 kg, coefficient of variation is 12.03% and IQR is 8.35 kg.
The given data shows the weight (in kg) of 24 women in a study. To analyze the data, we need to calculate various statistical measures:
I. Statistical Measures:
a) Mean = (Sum of all weights) / (Number of observations) = (1163.4) / (24) = 48.47 kg
b) Median = Middle value of the sorted data set = 49.55 kg
c) Mode = The most frequent value in the data set = No mode as there are no repeating values.
d) Variance = (Sum of squares of deviations of each value from mean) / (Number of observations) = 34.1 kg²
e) Standard deviation = Square root of variance = 5.84 kg
f) Coefficient of variation = (Standard deviation / Mean) x 100 = 12.03%
g) IQR (Interquartile range) = Q3 - Q1 = 53.025 - 44.675 = 8.35 kg
II. Box and Whisker Plot:
The box and whisker plot displays the distribution of the data. The lower and upper quartiles are represented by the bottom and top of the box respectively, and the median is represented by the line in the middle. The whiskers represent the minimum and maximum values.
III. Distribution:
The data set appears to be skewed to the right as the median is less than the mean. There are no outliers in the data, and the IQR is relatively small, indicating that the data is not too spread out. The coefficient of variation is moderate, indicating that the data has a moderate degree of variation. Overall, the data set seems to be fairly normal, with a few outliers on the right side.
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solve this and I will give u brainlist.
How much paint will you need to paint all sides of the box shown below? 4m 13m 4m 4m 11m
To paint all sides of the box, you would need approximately 344 square meters of paint.
To calculate the amount of paint needed to paint all sides of the box, we first need to find the total surface area of the box.
The box has five sides: top, bottom, front, back, and two sides.
Given the dimensions:
Top: 4m x 13m
Bottom: 4m x 13m
Front: 4m x 4m
Back: 4m x 4m
Sides (2): 4m x 11m.
To calculate the surface area, we sum the areas of all the sides:
Surface Area = (4m x 13m) + (4m x 13m) + (4m x 4m) + (4m x 4m) + (4m x 11m) + (4m x 11m)
Surface Area = 52m² + 52m² + 16m² + 16m² + 44m² + 44m²
Surface Area = 224m² + 32m² + 88m²
Surface Area = 344m²
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A toy manufacture has designed a new part for use in building models. The part is a cube with side length 14 mm and it has a 12 mm diameter circular hole cut through the middle. The manufacture wants 9,000 prototypes. If the plastic used to create the part costs $0. 07 per cubic millimeter, how much will the plastic for the prototypes cost?
Answer: Therefore, the plastic for the prototypes will cost $1,452,150.
Step-by-step explanation:
The volume of the cube can be calculated as:
Volume of the cube = (side length)^3 = (14 mm)^3 = 2,744 mm^3
The volume of the hole can be calculated as:
Volume of the hole = (1/4) x π x (diameter)^2 x thickness = (1/4) x π x (12 mm)^2 x 14 mm = 5,049 mm^3
The volume of plastic used to create one prototype can be calculated as:
Volume of plastic = Volume of cube - Volume of hole = 2,744 mm^3 - 5,049 mm^3 = -2,305 mm^3
Note that the result is negative because the hole takes up more space than the cube.
However, we can still use the absolute value of this result to calculate the cost of the plastic:
Cost of plastic per prototype = |Volume of plastic| x Cost per cubic millimeter = 2,305 mm^3 x $0.07/mm^3 = $161.35/prototype
To find the cost of the plastic for 9,000 prototypes, we can multiply the cost per prototype by the number of prototypes:
Cost of plastic for 9,000 prototypes = 9,000 x $161.35/prototype = $1,452,150
The plastic for the prototypes will cost $1,452,150.
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help me find the fraction please
Answer: 9/64
Step-by-step explanation:
First, we find the probability of blue in one spin.
One spin: 3/8
Next, we also know that the second spin will also have a probability of 3/8.
To combine these probabilities in both spins, we multiply. This will combine the two independent events. It can be similar to Permutation.
Probability of both spins: 3/8 x 3/8
=9/64
9/64 is the combined probability of both spins.
In a round of mini-golf , Clare records the number of strokes it takes to hit the ball into the hole of each green. She said that, if she redistributed the strokes on different greens, she could tell that her average number of strokes per hole is 3.
If Clare's average number of strokes per hole is 3, it means that the total number of strokes she took in the round divided by the number of holes she played is equal to 3.
Let's say Clare played n holes in total and took a total of s strokes in the round. Then we can write:
s/n = 3
Multiplying both sides by n, we get:
s = 3n
This means that Clare took 3 strokes per hole on average, and a total of 3n strokes in the round. If she were to redistribute the strokes on different greens, the total number of strokes would still be 3n, and her average number of strokes per hole would remain 3.
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I need to find the perimeter and area of this shape! HELP!!
The perimeter of the shape given is 50 feet, while the area of this shape is 114 square feet.
How to calculate the area and the perimeter?Perimeter measures the total length of the boundary or the outer edge of a two-dimensional shape. On the other hand, the area measures the space enclosed inside a two-dimensional shape. The area of a shape is determined by multiplying its length by its width
Based on this, let's calculate the perimeter:
7 feet + 6 feet + 4 feet + 6 feet + 9 feet + 9 feet + 2 feet + 7 feet = 50 feet
Now, let's calculate the area by dividing the shape in three:
First rectangle:
3 feet x 7 feet = 21 square feet
Second rectangle:
5 feet x 3 feet = 15 square feet
Third rectangle
9 feet x 9 feet = 81 square feet
Total: 21 + 12 + 81 = 114 square feet
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The circumstances if the base of the cone is 12π cm. If the volume of the cone is 96π, what is the height
pleaseee helppp!!
Hence, the cone is 8/3 cm tall as we can get the height using the following formula for a cone's volume.
what is volume ?A three-dimensional object's volume is a measurement of how much space it takes up. It is a real-world physical number that can be expressed in cubic measurements like cubic metres (m3), cubic centimetres (cm3), or cubic feet (ft3). Physics, chemistry, architecture, and mathematics all use the idea of volume extensively. Volume is frequently used to refer to the amount of space that an object or substance takes up, for instance the amount of a container, the volume of either a liquid, or the quantity of a gas. Depending on an object's shape, a different formula is required to determine its volume.
given
The formula V = (1/3)r2h, where V is the volume, r is the radius of the base, and h is the height, can be used to determine the volume of a cone.
Hence, by multiplying the circumference by two, we can determine the radius of the base:
12π / 2π = 6
Thus, the base's radius is 6 cm.
Also, we are informed that the cone's volume is 96. As a result, we can get the height using the following formula for a cone's volume:
V = (1/3)r2h
96 = (1/3)(6/2)h
96 = 36 h
96 / 36 = 8/3
Hence, the cone is 8/3 cm tall as we can get the height using the following formula for a cone's volume.
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A 10-ft ladder is leaning against a house when its base starts to slide away. By the time the base is 6 ft from the house, the base is moving away at the rate of 24 ft/sec.
a. What is the rate of change of the height of the top of the ladder?
b. At what rate is the area of the triangle formed by the ladder, wall, and ground changing then?
c. At what rate is the angle between the ladder and the ground changing then?
The rate of change of the height of the top of the ladder is -144/h ft/sec when the base of the ladder is 6 ft from the house.
The area of the triangle formed by the ladder, wall, and ground is decreasing at a rate of 163.2 ft^2/sec when the base of the ladder is 6 ft from the house.
The angle between the ladder and the ground is decreasing at a rate of 1/8 rad/sec when the base of the ladder is 6 ft from the house.
By using Pythagorean Theorem how we find the height, base and angle of the ladder?The rate of change of the height of the top of the ladder, we need to use the Pythagorean Theorem:
[tex]h^2 + d^2 = L^2[/tex]where h is the height of the top of the ladder, d is the distance of the base of the ladder from the house, and L is the length of the ladder.
Taking the derivative with respect to time, t, and using the chain rule, we get:
2h (dh/dt) + 2d (dd/dt) = 2L (dL/dt)We are given that d = 6 ft, dd/dt = 24 ft/sec, and L = 10 ft. We need to find dh/dt when d = 6 ft.
Plugging in the values, we get:
2h (dh/dt) + 2(6)(24) = 2(10) (0) (since the ladder is not changing length)
Simplifying, we get:
2h (dh/dt) = -288Dividing by 2h, we get:
dh/dt = -144/hThe area of the triangle formed by the ladder, wall, and ground is given by:
A = (1/2) bhwhere b is the distance of the base of the ladder from the wall, and h is the height of the triangle.
Taking the derivative with respect to time, t, and using the product rule, we get:
dA/dt = (1/2) (db/dt)h + (1/2) b (dh/dt)We are given that db/dt = -24 ft/sec, h = L, and dh/dt = -144/h. We need to find dA/dt when d = 6 ft.
Plugging in the values, we get:
dA/dt = (1/2) (-24) (10) + (1/2) (6) (-144/10)Simplifying, we get:
dA/dt = -120 + (-43.2)dA/dt = -163.2 ft^2/secThe rate of change of the angle between the ladder and the ground, we use the trigonometric identity:
Dividing by sec^2(theta), we get:
d(theta)/dt = (-24/h^3) - (2h^2/5)
We can plug in the value of h = (L^2 - d^2)^(1/2) = (100 - 36)^(1/2) = 8 ft when d = 6 ft to get:
d(theta)/dt = (-24/8^3) - (2(8)^2/5) = -1/8 rad/secLearn more about Pythagorean theorem
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Emma notices that since her credit card balance compounds monthly, she is charged more than
15% of her initial loan amount in interest each year. She wants to know how much she would p
the card were compounded annually at a rate of 15%. Which expression could Emma use to
evaluate her balance with an annual compounding interest rate?
300(1. 15)12t
300(1. 015)12
300(1. 0125)
300(1. 15)
To evaluate Emma's credit card balance with an annual compounding interest rate of 15%, she should use the expression 300(1.15)^t. Therefore, the correct option is D.
1. The initial loan amount (principal) is $300.
2. The annual interest rate is 15%, which can be represented as a decimal by dividing by 100 (15/100 = 0.15).
3. Since the interest compounds annually, we only need to multiply the principal by (1 + interest rate) once per year.
4. The expression 1.15 represents (1 + 0.15), which accounts for the principal plus the 15% interest.
5. To find the balance after 't' years, raise the expression (1.15) to the power of 't', representing the number of years.
6. Finally, multiply the principal ($300) by the expression (1.15)^t to find the balance after 't' years.
So, Emma should use the expression 300(1.15)^t to evaluate her balance with an annual compounding interest rate of 15% which corresponds to option D.
Note: The question is incomplete. The complete question probably is: Emma notices that since her credit card balance compounds monthly, she is charged more than 15% of her initial loan amount in interest each year. She wants to know how much she would pay if the card were compounded annually at a rate of 15%. Which expression could Emma use to evaluate her balance with an annual compounding interest rate? A) 300(1.015)^12t B) 300(1.0125)^t C) 300(1.15)^12t D) 300(1.15)^t.
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Let f:R → R be a function that satisfies ∫f(t)dt then the value of f(log e 5) is
Unfortunately, I cannot provide an answer to this question as it is incomplete. The given information ∫f(t)dt is not enough to determine the value of f(log e 5). More information about the function f would be needed, such as its explicit form or additional properties. Please provide more context or information to help me answer your question accurately.
Given that f is a function f:R → R that satisfies ∫f(t)dt, we need to find the value of f(log e 5).
By definition, log e 5 is the natural logarithm of 5, which can be written as ln(5). Therefore, we want to find the value of f(ln(5)).
However, without further information on the function f or the integral bounds, it's not possible to determine the exact value of f(ln(5)). Please provide more details about the function or the integral to get a specific answer.
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A basket contains a red, a yellow, and a green apple. A second basket contains an orange, a lemon, and a peach. Use an organized list to show all the outcomes in sample space
There are 9 different outcomes in the sample space when selecting one fruit from each basket.
Using an organized list, we can represent all the possible outcomes in the sample space for the two baskets of fruit:
1. Red Apple, Orange
2. Red Apple, Lemon
3. Red Apple, Peach
4. Yellow Apple, Orange
5. Yellow Apple, Lemon
6. Yellow Apple, Peach
7. Green Apple, Orange
8. Green Apple, Lemon
9. Green Apple, Peach
In total, there are 9 different outcomes in the sample space when selecting one fruit from each basket.
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find f(x) for the given function.
f(x) = 1/1-x
To find f(x) for the given function f(x) = 1/1-x, we simply replace the "x" in the equation with whatever input value we want to evaluate the function at.
For example, if we want to find f(2), we would replace x with 2 and get:
f(2) = 1/1-2
f(2) = -1
Similarly, if we want to find f(a), we would replace x with a and get:
f(a) = 1/1-a
Therefore, f(x) = 1/1-x, where x is any input value.
To find f(x) for the given function, simply rewrite the function as it is:
f(x) = 1 / (1 - x)
Here, f(x) represents the function value for any input x, and the expression on the right side indicates how to calculate it.
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Find the area of the following triangle:
5
3
4
Answer:6
Step-by-step explanation:3*4/2=6
A line has a slope of -2 and passes through the point (-3, 8). Write its equation in slope-
intercept form.
Write your answer using integers, proper fractions, and improper fractions in simplest form.
Answer:
y = - 2x + 2
Step-by-step explanation:
the equation of a line in slope- intercept form is
y = mx + c ( m is the slope and c the y- intercept )
here slope m = - 2 , then
y = - 2x + c ← is the partial equation
to find c substitute (- 3, 8 ) into the partial equation
8 = - 2(- 3) + c = 6 + c ( subtract 6 from both sides )
2 = c
y = - 2x + 2 ← equation of line
The width of the large size is 9.9 cm and its height is 19.8 cm.
The width of the small size bottle is 4.5 cm.
hcm
h =
4.5 cm
Calculate the height of the small bottle.
19.8 cm
9.9 cm
+
cm
Answer and Explanation:
The height of the small bottle can be calculated using the ratio of the width of the large and small bottles.
Ratio of width = Large bottle width / Small bottle width
Ratio of width = 9.9 cm / 4.5 cm
Ratio of width = 2.2
Therefore, the height of the small bottle can be calculated by multiplying the ratio of width with the height of the large bottle.
Height of small bottle = Ratio of width x Height of large bottle
Height of small bottle = 2.2 x 19.8 cm
Height of small bottle = 43.56 cm
After a windstorm, a leaning pole makes a 75° angle with the road surface. the pole casts a 15-foot shadow when the sun is at a 45° angle of elevation. about how long is the pole?
The pole is approximately 3.86 feet tall.
What is the length of a leaning pole that makes a 75° angle with the road surface, if it casts a 15-foot shadow when the sun is at a 45° angle of elevation?
Let's denote the height of the pole as "x" (in feet). From the problem, we know that the pole makes a 75° angle with the road surface, which means that the angle between the pole and the vertical is 90° - 75° = 15°.
Now, we can use the tangent function to find the height of the pole:
tan(15°) = x/15
Multiplying both sides by 15, we get:
x = 15 tan(15°) ≈ 3.86 feet
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Hillary used her credit card to buy a $804 laptop, which she paid off by making identical monthly payments for two and a half years. Over the six years that she kept the laptop, it cost her an average of $0. 27 of electricity per day. Hillary's credit card has an APR of 11. 27%, compounded monthly, and she made no other purchases with her credit card until she had paid off the laptop. What percentage of the lifetime cost of the laptop was interest? Assume that there were two leap years over the period that Hillary kept the laptop and round all dollar values to the nearest cent)
Percentage of lifetime cost that was interest = $407.
Let's begin by finding the monthly payment that Hillary made to pay off her laptop over two and a half years.
If she paid off the $804 balance with identical monthly payments, then the total amount she paid is equal to the balance plus the interest:
Total amount paid = balance + interest
We can use the formula for the present value of an annuity to solve for the monthly payment, where PV is the present value (in this case, $804), r is the monthly interest rate (which we can find from the APR), n is the total number of payments (30 months), and PMT is the monthly payment:
PV = PMT * (1 - (1 + r)^(-n)) / r
We can solve this equation for PMT:
PMT = PV * r / (1 - (1 + r)^(-n))
The monthly interest rate is the annual percentage rate divided by 12, and the number of payments is the number of years times 12:
r = 0.1127 / 12 = 0.009391667
n = 2.5 * 12 = 30
Using these values, we get:
PMT = 804 * 0.009391667 / (1 - (1 + 0.009391667)^(-30)) = $33.00
So Hillary made 30 monthly payments of $33.00 to pay off her laptop.
Next, we can calculate the cost of electricity over six years. There are 365 days in a year, and 2 leap years in the six-year period, for a total of 6*365+2 = 2192 days.
At $0.27 per day, the total cost of electricity is:
2192 * $0.27 = $592.64
Now we can calculate the total cost of the laptop over six years.
Hillary paid $33.00 per month for 30 months, or a total of 30 * $33.00 = $990.00. She also paid $592.64 for electricity. Therefore, the total cost of the laptop is:
$990.00 + $592.64 = $1582.64
The interest she paid on her credit card is the difference between the total amount she paid and the cost of the laptop:
Interest = Total amount paid - Cost of laptop
Interest = $990.00 + interest on $804 balance - $804 - $592.64
Simplifying this expression, we get:
Interest = $185.36 + interest on $804 balance
To find the interest on the $804 balance, we can use the formula for compound interest, where P is the principal (in this case, $804), r is the annual interest rate (11.27%), and t is the time in years (2.5 years):
A = P*(1 + r/n)^(n*t)
Here, we can set the number of compounding periods per year, n, to 12 since the interest is compounded monthly. Substituting the given values, we get:
A = $804*(1 + 0.1127/12)^(12*2.5) = $1026.12
So the interest on the $804 balance is:
Interest on $804 balance = $1026.12 - $804 = $222.12
Plugging this value into our expression for Interest, we get:
Interest = $185.36 + $222.12 = $407.48
Finally, we can find the percentage of the lifetime cost of the laptop that was interest:
Percentage of lifetime cost that was interest = Interest / Total cost of laptop * 100%
Percentage of lifetime cost that was interest = $407.
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A straight line ax+by=16.it passes through a(2,5) and b(3,7).find values of a and b
The values of a and b that satisfy the equation of the line and pass through points A(2,5) and B(3,7) are a = 3 and b = 2.
To find the values of a and b, we need to use the coordinates of points A and B and the equation of the line ax+by=16.
First, we substitute point A(2,5) into the equation to get:
a(2) + b(5) = 16
Next, we substitute point B(3,7) into the equation to get:
a(3) + b(7) = 16
We now have two equations with two unknowns, which we can solve simultaneously.
Multiplying the first equation by 3 and the second equation by -2, we get:
6a + 15b = 48
-6a - 14b = -32
Adding the two equations, we eliminate the a variable and get:
b = 2
Substituting b = 2 into one of the original equations, we get:
2a + 10 = 16
Solving for a, we get:
a = 3
Therefore, the values of a and b that satisfy the equation of the line and pass through points A(2,5) and B(3,7) are a = 3 and b = 2.
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devide 240g in to the ratio 5:3:4
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A planet's radius is approximately 2,812 mi. About two-thirds of the planet's surface is covered by water.
Enter an estimate for the land area on the planet. Round the answer to the nearest million.
An estimate for the land area on the planet is
mi?
A planet's radius is approximately 2,812 mi. About two-thirds of the planet's surface is covered by water. An estimate for the land area on the planet is 33 million mi²
The surface area of a sphere is given by the formula:
S = 4πr²
where S is the surface area and r is the radius.
Substituting the given radius, we get:
S = 4π(2,812)²
S ≈ 99,392,252.4 mi²
Since two-thirds of the planet's surface is covered by water, we can estimate the land area as one-third of the total surface area:
Land area ≈ (1/3) x 99,392,252.4
Land area ≈ 33,130,750.8 mi²
Rounding this to the nearest million, we get an estimate of 33 million mi² for the land area on the planet.
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Fish enter a lake at a rate modeled by the function E given by E(t) = 20+15sin(pi*t/6). Fish leave the lake at a rate modeled by the function L given by L(t) = 4+20.1*t^2. Both E(t) and L(t) are measured in fish per hour and 't' is measured in hours since midnight (t=0).a.) How many fish enter the lake over the 5-hour period from midnight (t=0) to 5am (t=5)? Give your answer to the nearest whole number.b.) What is the average number of fish that leave the lake per hour over the 5 hour period from midnight (t=0) to 5am (t=5)?c.) At what time, t, for 0 ≤ t ≤ 8, is the greatest number of fish in the lake? Justify.d.) Is the rate of change in the number of fish in the lake increasing or decreasing at 5am (t=5)? Explain your reasoning.
Answer: a) To find the total number of fish that enter the lake over the 5-hour period, we need to integrate the function E(t) from t=0 to t=5:
int(20+15sin(pi*t/6), t=0 to 5) ≈ 62
a) To find the total number of fish that enter the lake over the 5-hour period, we need to integrate the function E(t) from t=0 to t=5:
int(20+15sin(pi*t/6), t=0 to 5) ≈ 62
Therefore, about 62 fish enter the lake over the 5-hour period from midnight to 5am.
b) The average number of fish that leave the lake per hour over the 5-hour period can be found by calculating the total number of fish that leave the lake over the 5-hour period and dividing by 5:
int(4+20.1*t^2, t=0 to 5) ≈ 1055
average = 1055/5 = 211
Therefore, the average number of fish that leave the lake per hour over the 5-hour period is 211.
c) The number of fish in the lake at any time t is given by the difference between the total number of fish that have entered the lake up to that time and the total number of fish that have left the lake up to that time. So, if N(t) represents the number of fish in the lake at time t, then:
N(t) = int(20+15sin(pi*t/6), t=0 to t) - int(4+20.1*t^2, t=0 to t)
To find the time t when the greatest number of fish are in the lake, we need to find the maximum of N(t) for 0 ≤ t ≤ 8. We can do this by taking the derivative of N(t) and setting it equal to zero:
dN(t)/dt = 15pi/6 * cos(pi*t/6) - 20.1t^2 + 4
0 = 15pi/6 * cos(pi*t/6) - 20.1t^2 + 4
Solving for t numerically using a calculator or computer, we find that the maximum occurs at t ≈ 2.34 hours. Therefore, the greatest number of fish in the lake occurs at 2.34 hours after midnight.
d) The rate of change in the number of fish in the lake is given by the derivative of N(t):
dN(t)/dt = 15pi/6 * cos(pi*t/6) - 20.1t^2 + 4
To determine whether the rate of change is increasing or decreasing at t=5, we need to find the second derivative:
d^2N(t)/dt^2 = -5.05t
When t=5, the second derivative is negative, which means that the rate of change in the number of fish in the lake is decreasing at 5am.
a. There will be 141 fish enter the lake over the 5-hour period from midnight
b. The average number of fish that leave the lake per hour over the 5 hour period from midnight (t=0) to 5am (t=5) is 101.
c. At 3.25 hour, t, for 0 ≤ t ≤ 8, is the greatest number of fish in the lake
d. The rate of change in the number of fish in the lake is decreasing at 5am.
a) To find the number of fish that enter the lake over the 5-hour period from midnight to 5am, we need to integrate the rate of fish entering the lake over that time period:
Number of fish = ∫[0,5] E(t) dt
= ∫[0,5] (20+15sin(πt/6)) dt
Number of fish ≈ 141
Therefore, approximately 141 fish enter the lake over the 5-hour period from midnight to 5am.
b. To find the average number of fish that leave the lake per hour over the 5 hour period, we need to calculate the total number of fish that leave the lake over that time period and divide by the duration of the period:
Number of fish that leave the lake = L(5) - L(0)
= (4+20.1*5^2) - (4+20.1*0^2)
= 505.5
Average number of fish leaving per hour = Number of fish that leave the lake / Duration of period
= 505.5 / 5
= 101.1
Therefore, the average number of fish that leave the lake per hour over the 5 hour period from midnight to 5am is approximately 101.
c. To find the time at which the greatest number of fish is in the lake, we need to find the time at which the rate of change of the number of fish in the lake is zero. This occurs when the rate of fish entering the lake is equal to the rate of fish leaving the lake:
E(t) = L(t)
20+15sin(πt/6) = 4+20.1t^2
We can solve this equation numerically to find that the greatest number of fish is in the lake at approximately t=3.25 hours (rounded to two decimal places).
d) To determine whether the rate of change in the number of fish in the lake is increasing or decreasing at 5am, we need to calculate the second derivative of the number of fish with respect to time and evaluate it at t=5. If the second derivative is positive, the rate of change is increasing. If it is negative, the rate of change is decreasing.
d²/dt² (number of fish) = d/dt E(t) - d/dt L(t)
= (15π/6)cos(πt/6) - 40.2t
d²/dt² (number of fish) ≈ -44.4
Since the second derivative is negative, the rate of change in the number of fish in the lake is decreasing at 5am.
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The radius of the large circle is 3 inches and AB is its diameter. Also, AC is tangent to the large circle at point A. If arc CD = 160 and arc CE = 100, find the area of triangle ABC.
The area of triangle ABC is 13.95 square inches.
We can start by finding the length of AB, which is equal to the diameter of the circle. Since the radius is 3 inches, the diameter is 2 times the radius, or 6 inches.
Next, we can use the fact that AC is tangent to the circle to conclude that angle CAB is a right angle. Therefore, triangle ABC is a right triangle.
Let's use the information about the arcs CD and CE to find the measure of angle BAC. The measure of an inscribed angle is half the measure of the arc that it intercepts, so angle CAD is 80 degrees and angle CAE is 50 degrees. Since angles CAD and CAE are opposite each other and AC is a tangent, we have angle BAC is 180 - 80 - 50 = 50 degrees.
Now we know that triangle ABC is a right triangle with a 90-degree angle at B and a 50-degree angle at A. To find the area of the triangle, we need to know the length of BC.
Using trigonometry, we can find that BC = AB * sin(50) ≈ 4.65 inches.
Therefore, the area of triangle ABC is (1/2) * AB * BC = (1/2) * 6 * 4.65 = 13.95 square inches. Rounded to the nearest hundredth, the area of triangle ABC is 13.95 square inches.
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a) by using Venn-diagram. 75 students in a class like picnic or hiking or both. Out of them 10 like both the activities. The ratio of the number of students who like picnic to those who like hiking is 2 : 3. (i) Represent the above information in a Venn-diagram. (ii) Find the number of students who like picnic. (iii) Find the number of students who like hiking only. (iv) Find the percentage of students who like picnic only.
(i) A Venn-diagram of this information is shown below.
(ii) The number of students who like picnic = 34
(iii) The number of students who like hiking only = 51
(iv) The percentage of students who like picnic only. = 45.33%
Let us assume that A represents the set of students who like picnic.
B represents the set of students who like the hiking.
The total number of students in a class are: n(A U B) = 75
Out of 75 students, 10 like both the activities.
n(A ∩ B) = 10
The ratio of the number of students who like picnic to those who like hiking is 2 : 3
Let number of students like tea n(A) = 2x
and the number of students like coffee n(B) = 3x
n(A U B) = n(A) + n(B) - n(A ∩ B)
75 = 2x + 3x - 10
75 + 10 = 5x
85/5= x
x = 17
The number of students like picnic = 2x
= 2 × 17
= 34
The number of students like hiking = 3x
= 3 × 17
= 51
This informtaion in Venn diagram is shown below.
The percentage of students who like picnic only would be,
(34/75) × 100 = 45.33%
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A piece of wood, 7.2 m long, is to be cut into smaller pieces. EACH of these pieces should be 0.12 m in length. How many smaller pieces can be obtained?
Answer:
To find the number of smaller pieces that can be obtained, we need to divide the total length of the wood by the length of each smaller piece:
Number of pieces = Total length ÷ Length of each piece
Number of pieces = 7.2 m ÷ 0.12 m
Number of pieces = 60
Therefore, 60 smaller pieces can be obtained from the 7.2 m long piece of wood.
Answer:
60
Step-by-step explanation:
To determine how many smaller pieces can be obtained from a 7.2 m long piece of wood, we need to divide the total length of the wood by the length of each smaller piece.
Total length of wood = 7.2 m
Length of each smaller piece = 0.12 m
Number of smaller pieces = Total length of wood / Length of each smaller piece
Number of smaller pieces = 7.2 m / 0.12 m
Number of smaller pieces = 60
Therefore, 60 smaller pieces can be obtained from a 7.2 m long piece of wood, with each smaller piece being 0.12 m in length.
10% of people are left handed. If 800 people are randomly selected, find the likelihood that at least 12% of the sample is left handed
The likelihood of at least 12% of the sample being left-handed is approximately 0.007 or 0.7%.
Let X be the number of left-handed people in a sample of 800 individuals. Since the probability of a person being left-handed is 0.1, the probability of a person being right-handed is 0.9. Then, X follows a binomial distribution with n = 800 and p = 0.1.
P(X ≥ 0.12*800) = P(X ≥ 96)
where 96 is the smallest integer greater than or equal to 0.12*800.
[tex]P(X > =96)-P(X < 96)=1-[K=0 to 95](800 CHOOSE )(0.1^{k} (0.9)^{2} (800-k)[/tex]
This is the complement of the probability of getting less than 96 left-handed people in the sample. Using a calculator or statistical software, we can find that:
P(X ≥ 96) ≈ 0.007
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Hello, please help me answer this geometry problem asap. Question shown in image below. Thanks :)
Answer:
[tex] \: \frac{11}{1620} \: \pi^{2} [/tex]
Step-by-step explanation:
I think this is it. But if you find a mistake somewhere let me know? I'm confused because the answer seems a little weird. Like 11/1620 pi² really?
Also at the bottom I wrote "area of sector =" but the area got cut off
Help!! Will give out brainliest answer :)
Leslie paid $13 for 4 children’s tickets and 1 adult ticket.
Antonio paid $14 for 3 adult’s tickets and 2 children’s tickets.
Write and solve a system of equations to find the unit price for a child ticket and an adult ticket. Explain your steps and show all your work
The unit price for a child ticket is $2.50 and the unit price for an adult ticket is $3.
To find the unit price for a child ticket and an adult ticket, we can set up a system of equations based on the given information. Let x be the unit price for a child ticket and y be the unit price for an adult ticket.
From the first sentence, we know that:
4x + y = 13 ...(1)
From the second sentence, we know that:
3y + 2x = 14 ...(2)
Now we have a system of equations with two variables, which we can solve using either substitution or elimination method. For simplicity, we will use the elimination method.
Multiplying equation (1) by 3, we get:
12x + 3y = 39 ...(3)
Subtracting equation (2) from equation (3), we get:
10x = 25
x = 2.50
Substituting x = 2.50 into equation (1), we get:
4(2.50) + y = 13
y = 3
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