The probability that the mean yield of the 80 one acre plots chosen will be within 2 bushels of the population mean is 0.9332. The assumptions made to answer part (a) are that the population standard deviation is known, the sample is randomly selected, and the sample size is large (n>=30)
a) We can use the central limit theorem to approximate the distribution of the sample mean. The distribution of the sample mean can be approximated as a normal distribution with a mean of 107 bushels and a standard deviation of 12/√80=1.34 bushels. Therefore, we can find the probability that the sample mean falls within 2 bushels of the population mean using the standard normal distribution.
P(105 ≤ X ≤ 109) = P(Z ≤ (109-107)/1.34) - P(Z ≤ (105-107)/1.34)
= P(Z ≤ 1.49) - P(Z ≤ -1.49)
= 0.9319 - 0.0681
= 0.8638
Therefore, there is an 86.38% probability that the mean yield of the 80 one acre plots will be within 2 bushels of the population mean.
b) Assumptions made:
The sample of 80 one acre plots is selected at random.
The one acre plots are independent of each other.
The population standard deviation is known and remains constant.
The distribution of the yield of corn per acre is approximately normal.
The sample size is large enough (n≥30) to apply the central limit theorem.
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john and jane are married. the probability that john watches a certain television show is .5. the probability that jane watches the show is .3. the probability that john watches the show, given that jane does, is .5. (a) find the probability that both john and jane watch the show. (round your answer to 2 decimal places.) (b) find the probability that jane watches the show, given that john does. (round your answer to 3 decimal places.) (c) do john and jane watch the show independently of each other? multiple choice yes no
(a) The probability that both John and Jane watch the show is 0.15 or 15% (rounded to 2 decimal places).
(b)The probability that Jane watches the show, given that John does, is 0.3 or 3.
(c) Since these probabilities are equal, John and Jane watch the show independently of each other.
The answer is Yes.
John and Jane watch the show independently of each other if
(a) To find the probability that both John and Jane watch the show, we can use the conditional probability formula:
P(A and B) = P(A|B) * P(B),
where A represents John watching the show and B represents Jane watching the show.
Given, P(A|B) = 0.5 and P(B) = 0.3.
P(A and B) = 0.5 * 0.3 = 0.15.
Therefore, the probability that both John and Jane watch the show is 0.15 or 15% (rounded to 2 decimal places).
(b) To find the probability that Jane watches the show, given that John does, we can use the conditional probability formula: P(B|A) = P(A and B) / P(A).
Given, P(A and B) = 0.15 and P(A) = 0.5.
P(B|A) = 0.15 / 0.5 = 0.3.
Therefore, the probability that Jane watches the show, given that John does, is 0.3 or 30% (rounded to 3 decimal places).
(c) John and Jane watch the show independently of each other if P(A and B) = P(A) * P(B).
In this case, P(A and B) = 0.15 and P(A) * P(B) = 0.5 * 0.3 = 0.15.
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pls answer me this asap, i will do anything
Answer:
a 2.009, 2.15, 2.7
b 3.2, 3.342, 3.45
c 17.05, 17.1, 17.125, 17.42
Solve each question show solutions on the number line |x+8|=-2
On the number line, the empty set, denoted by an empty interval or a pair of crossed-out points, would be the solution set.
what is equation ?A mathematical assertion that establishes the equal of two expressions is called an equation. Variables, can serve as placeholders with unknowable values, and multipliers, and that are constants can multiply the variables, are frequently included. Determine you significance of the variables necessary make the answer true by solving the equation. The significant early (=), plus sign (+), and debit sign (-), among several other mathematical symbols, can be used to indicate equations. They are employed in many disciplines, such as mathematics, mathematics, chemistry, construction, and economics, to model the interactions between variables and to address issues.
given
Due to the fact that a real number's absolute value is never negative, the equation |x+8|=-2 has no solutions.
As a result, it is impossible for x+8 to have an absolute value of -2, which is a negative number.
On the number line, the empty set, denoted by an empty interval or a pair of crossed-out points, would be the solution set.
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Three dogs can eat 3 bones in 3 days. If this efficiency is followed, how many bones can 9 dogs eat in 9 days?
Answer:
If three dogs can eat three bones in three days, we can assume that each dog eats one bone in three days. Therefore, in one day, one dog can eat one-third of a bone.
If nine dogs were to eat in the same efficiency, we can multiply the number of dogs by the fraction of bones each can eat per day. So, nine dogs can eat 3 times more than 3 dogs in a day, which is 3 x 3 = 9 bones. In one day, each dog can eat one-third of the bone, and therefore nine dogs will eat 9 x 1/3 = 3 bones.
This means that in nine days, nine dogs can eat 9 x 3 = 27 bones.
The Area of a Rectangle is 56. The width is 2 more than 3
times the length. Find the width.
O 15
O 14
08
06
Answer:
14
Step-by-step explanation:
A = length x width
l = length
w = 3l + 2
A = l(3l + 2) = 56
3l² + 2l - 56 = 0
Now you have a polynomial you can solve by factoring or using the quadratic equation to find the 2 roots of l:
using the quadratic equation with a = 3, b = 2, c = -56
l = 4, -14/3
Disregarding the negative root,
l = 4
Therefore, w = 3(4) + 2 = 14
please find the answer
the equation y=1 does not represent a direct variation, whereas the other three equations given do represent direct variation.
What is Direct variation?
Direct variation is a relationship between two variables in which one is a constant multiple of the other. This means that if one variable increases, the other variable will also increase by a proportional amount. In other words, direct variation can be represented by the equation y=kx, where k is the constant of variation.
Out of the four equations given, the equation y=1 does not represent a direct variation. This is because the equation is not a function of x. In other words, no matter what value of x we choose, y will always be equal to 1. This means that the two variables, x and y, are not related to each other in any way. Therefore, there is no constant k that can be multiplied by x to get y.
On the other hand, the other three equations do represent direct variation. For example, in the equation y=x, y is equal to x multiplied by the constant k=1. Similarly, in the equation y=-5x, y is equal to x multiplied by the constant k=-5. Finally, in the equation y=0.9x, y is equal to x multiplied by the constant k=0.9.
In summary, the equation y=1 does not represent a direct variation, whereas the other three equations given do represent direct variation.
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What is the greatest common factor of 42 and 56?
Answer: 14
Step-by-step explanation: you need to start by finding the factors of them and then what ever the biggest number is in both of them has to match but i got 14.
Anyone help in number 5?
Answer:
Step-by-step explanation:
5)
Draw a right triangle with a hypotenuse of 60 km, a base angle of 60°, and the base leg of x.
Find x, the distance it traveled to the east:
cos60° = x/60
x = 60(cos60) = 30 km
~~BRAINILY~~
The functions f(x) = x2 – 2 and g(x) = –x2 + 5 are shown on the graph.
To graph the solution set of the system of inequalities y>x²-2 and y ≥ -x²+5, which is the area between the two curves, above the curve of f(x) = [tex]x^2 - 2[/tex]and above or on the curve of g(x) = [tex]-x^2 + 5[/tex].
To graph the solution set to the system of inequalities y>x²-2 and y ≥ -x²+5, you can follow these steps:
Graph the functions f(x) = x² - 2 and g(x) = -x² + 5 on the same coordinate plane.
Shade the region above the curve of f(x) = x² - 2 since the inequality y > x² - 2 indicates that the values of y are greater than the corresponding values of x² - 2.
Shade the region above or on the curve of g(x) = -x² + 5 since the inequality y ≥ -x²+5 indicates that the values of y are greater than or equal to the corresponding values of -x² + 5. This region can be represented as the area bounded by the curve of g(x) and the x-axis.
This shaded region is the area between the two curves, above the curve of f(x) = x² - 2, and above or on the curve of g(x) = -x² + 5. This region can be described as the set of all points (x, y) in the coordinate plane that satisfy the two inequalities simultaneously.
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The functions f(x) = x2 – 2 and g(x) = –x2 + 5 are shown on the graph.
Explain how to modify the graphs of f(x) and g(x) to graph the solution set to the following system of inequalities How can the solution set be identified?
y>x²-2
y ≥ -x²+5
Find the distance, d, of AB.
A = (-2,-10) B = (-6,0)
-12-10 - B
A
-2
-B
-10
d = √(x2-x₁)² + (y2 - Y₁)²
d = [?]
Round to the nearest tenth.
Help Re
Skip
The calculated value of the distance between points A and B is 2√29 units.
Finding the distance, d, of AB.We can use the distance formula to find the distance between points A and B:
d = √[(x₂ - x₁)² + (y₂ - y₁)²]
where (x₁, y₁) = A = (-2, -10) and (x₂, y₂) = B = (-6, 0).
Substituting these values into the formula, we get:
d = √[(-6 - (-2))² + (0 - (-10))²]
d = √[(-4)² + 10²]
d = √(16 + 100)
d = √116
d = 2√29
Therefore, the distance between points A and B is 2√29 units.
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Leon is interested in the relationships among geographic region and
political affiliation. Leon collected data about the political parties of US
senators at that time and the regions of the US that they represent. His
results are in the table below.
His results are in the table are-
Southeast = 36.4%
Southwest = 9.1%
West = 18.2%
Midwest = 20%
What is arithmetic?
Arithmetic is an elementary part of mathematics that consists of the study of the properties of the traditional operations on numbers—addition, subtraction, multiplication, division, exponentiation, and extraction of roots.
Here, we have
: Leon is interested in the relationships between geographic region and political affiliation. Leon collected data about the political parties of US senators at that time and the regions of the US that they represent.
Southeast = 20/55 = 36.4%
Southwest = 5/55 = 9.1%
West = 10/55 = 18.2%
Midwest = 11/55 = 20%
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Determine the value of x
The value of variable x=4 units in the given figure.
Define right triangleA right triangle is a type of triangle that has one angle measuring exactly 90 degrees. The side opposite to the right angle is called the hypotenuse, while the other two sides are called the legs or catheti. The lengths of the legs and the hypotenuse of a right triangle are related by the Pythagorean theorem, which states that the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the legs.
From the given figure;
Tan60°=p/1
p=1.732 units
The smaller triangle is right angle
h²=p²+b²
h=√1²+1.732²
h=2units
In the bigger triangle, angle subtended by base is 60 (vertically opposite angle)
Using the trigonometric function
Cos60°=2/x
1/2=2/x
x=4units
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The graph of a quadratic function f is shown on the grid. The coordinates of the y-intercept and the vertex are integers.
Choose the correct answer from each drop-down menu to complete the statement.
The function has a ____________(Choose one Minimum or Maximum) value of __________.( Choose one -3, 0, 1 , 2)
The quadratic function f has a Minimum value of -3 at x = 2, taken from the given graph.
Explain about the minima for quadratic function:A parabola, a U-shaped curve, is the shape of a quadratic function's graph. The graph's vertex, which is an extreme point, is one of its key characteristics.
The vertex, or lowest point on the graph or minimal value of a quadratic function, is where the parabola will open up. The vertex is the highest set of points or the maximum value if the parabola opens downward. The vertex is a pivotal location on the graph in both scenarios. The graph is indeed symmetric, with the axis of symmetry being a vertical line that passes through the vertex.Given data:
On the grid, the quadratic function f has graph is displayed. The vertex and y-intercept have integer coordinates.As, the U shaped graph opens upwards, it has the minimum value at the turning point.
Thus, the quadratic function f has a Minimum value of -3 at x = 2, taken from the given graph.
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Amy will deposit $5,000 into a bank account paying an annual interest rate of 31%. What is the difference in the account balance between a simple interest applied or a compound interest applied over an 8-year time period, rounded to the nearest cent?
The difference in the account balance between a simple interest and a compound interest over an 8-year time period, rounded to the nearest cent, is $12,189.02.
To calculate the difference in the account balance between simple interest and compound interest over 8 years, we first need to determine the final balance for each case.
Simple Interest:
Simple interest is calculated as a percentage of the initial principal (or the original amount deposited) and is paid only on the principal amount. The formula for simple interest is:
I = P * r * t
where I is the interest earned, P is the principal amount (in this case, $5,000), r is the annual interest rate (as a decimal), and t is the time period in years.
For this problem, the interest rate is given as 31%, which is equivalent to 0.31 as a decimal. Therefore, we have:
I = $5,000 * 0.31 * 8 = $12,400
The final balance for simple interest is the sum of the principal and the interest earned:
Final balance (simple interest) = $5,000 + $12,400 = $17,400
Compound Interest:
Compound interest is interest that is calculated on both the principal amount and any accumulated interest. The formula for compound interest is:
A = P * (1 + r/n)^(n*t)
where A is the final amount, P is the principal amount, r is the annual interest rate (as a decimal), n is the number of times per year that interest is compounded, and t is the time period in years.
For this problem, the interest rate is 31% and the account is compounded annually, so n = 1. Therefore, we have:
A = $5,000 * (1 + 0.31/1)^(1*8) = $29,589.02
The final balance for compound interest is the amount we just calculated:
Final balance (compound interest) = $29,589.02
The difference in the two account balances is:
$29,589.02 - $17,400 = $12,189.02
Therefore, the difference in the account balance between a simple interest and a compound interest over an 8-year time period, rounded to the nearest cent, is $12,189.02.
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in july 2005, the internet was linked by a global network of about 352.1 million host computers. the number of host computers has been growing approximately exponentially and was about 35.3 million in july 1998. (a) find a formula for the number, n, of internet host computers (in millions of computers) as an exponential function of t, the number of years since july 1998, using exponential function of the form . what are the values of a and k in your model?
Answer:
a=671000
Step-by-step explanation:
woah
The formula for the number of internet host computers as an exponential function of t is: n(t) = 35.3 × e^((1/7) × ln(n(7) / 35.3) × t) and the values of a and k in the model are k = (1/7) × ln(n(7) / n(0)) and a = n(0) = 35.3 million
a. To find the values of a and k, we can use the information given in the question. Let n(0) be the number of host computers in July 1998, which is 35.3 million. Let n(7) be the number of host computers in July 2005, which is 352.1 million. We can use these values to solve for a and k.
n(0) = a × e^(k0) = a
n(7) = a × e^(k7)
Dividing the second equation by the first equation, we get:
n(7) / n(0) = e^(k×7)
Taking the natural logarithm of both sides, we get:
ln(n(7) / n(0)) = k×7
Therefore, k = (1/7) × ln(n(7) / n(0)) and a = n(0) = 35.3 million.
Therefore, the formula for the number of internet host computers as an exponential function of t is:
n(t) = 35.3 × e^((1/7) × ln(n(7) / 35.3) × t)
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A queen bee has a colony of 2,000 drones and 18,000 worker bees.6,000 of the worker bee forage for pollen and nectar. What part of the worker bees forage for pollen and nectar?
One-third of the worker bees forage for pollen and nectar.
What is percentage?
Percentage is a way of expressing a number as a fraction of 100. It is often denoted by the symbol "%". For example, 50% means "50 out of 100" or "50/100" or "0.5 as a decimal". Percentages are commonly used to express the proportion or relative size of a quantity with respect to another quantity. They can be used in a wide range of contexts, including finance, statistics, science, and everyday life.
Here out of the 18,000 worker bees, 6,000 forage for pollen and nectar.
To calculate the part of worker bees that forage for pollen and nectar, we can use the following formula:
Part = (Number of bees foraging / Total number of worker bees) x 100
Plugging in the numbers, we get:
Part = (6,000 / 18,000) x 100
Part = 0.333 x 100
Part = 33.3%
Therefore, 33.3% or approximately one-third of the worker bees forage for pollen and nectar.
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Help please need anwser
Answer:
1)55 and for 2) 35
Step-by-step explanation:
All you have to do is and for perimeter.
(17 POINTS) Which number should be added to both sides of this quadratic equation to complete the square?
Answer:
(b/2)² + 1 = x² - 3x + (b/2)²
b²/4 + 1 = x² - 3x + b²/4
b²/4 - b²/4 = x² - 3x - 1
x² - 3x - 1 = 0
x² - 3x = 1
x² - 3x + 9/4 = 1 + 9/4
(x - 3/2)² = 13/4
x - 3/2 = ±(√13/4)
x = ±√13/2 + 3/2
so the number that must be added will be 3/2
Rewrite the expression in terms of sine and cosine and utilize the Fundamental Pythagorean Identity: sin²(x)+cos²(x)=1
Verify the identity using the Pythagorean Identity:
[tex]\frac{1+csc(x)}{cot(x)+cos(x)}=sec(x)[/tex]
We have verified the identity of the expression 1 + cos(x) / cot(x) + cos(x) = sec(x) using the Pythagorean Identity.
What is a trigonometry?The mathematical subject of trigonometry is the study of the connections between the angles and sides of triangles.
let's rewrite the expression using sine and cosine:
1 + cos(x) / cot(x) + cos(x) = sec(x)
cot(x) is the same as cos(x)/sin(x), so we can substitute:
1 + cos(x) / (cos(x)/sin(x)) + cos(x) = sec(x)
Simplifying the expression in the denominator:
1 + cos(x) * (sin(x)/cos(x)) + cos(x) = sec(x)
Canceling out the cos(x) terms in the numerator:
1 + sin(x) + cos(x) = sec(x)
Now, let's use the Pythagorean Identity:
sin²(x) + cos²(x) = 1
Divide both sides by cos²(x):
tan²(x) + 1 = sec²(x)
Substitute tan(x) for sin(x)/cos(x):
(sin²(x)/cos²(x)) + 1 = sec²(x)
Multiply both sides by cos²(x):
sin²(x) + cos²(x) = cos²(x)sec²(x)
Substitute cos²(x) with 1 - sin²(x):
sin²(x) + (1 - sin²(x)) = (1 - sin²(x))sec²(x)
Simplify:
1 = sec²(x)
Taking the square root of both sides:
1 = sec(x)
Therefore, the identity is verified.
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After the cataract took Euler's sight, he
Please someone help me I need to find the answer and the solution
Answer:
12.7
Step-by-step explanation:
3x+10x-15=180
13x=165
x=12.69.....
x=12.7
A rocket is shot off from a launcher. The accompanying table represents the height of the rocket at given times, where x is time, in seconds, and y is height, in feet. Write a quadratic regression equation for this set of data, rounding all coefficients to the nearest tenth. Using this equation, find the height, to the nearest foot, at a time of 7.8 seconds.
1.1 308
1.8 489
3.1 761
3.7 891
4.8 1070
The height of the rocket at a time of 7.8 seconds is approximately 2,254 feet.
To find a quadratic regression equation, we need to fit a quadratic function of the form [tex]y = ax^2 + bx + c[/tex]to the data.
Using a calculator or spreadsheet software, we can find the coefficients that minimize the sum of the squared errors between the predicted values of y and the actual values of y.
The resulting quadratic regression equation is:
[tex]y = 88.6x^2 + 2.9x + 196.5[/tex]
To find the height of the rocket at a time of 7.8 seconds, we simply substitute x = 7.8 into the equation and evaluate:
[tex]y = 88.6(7.8)^2 + 2.9(7.8) + 196.5[/tex]
y ≈ 2,253.6
Rounding to the nearest foot, we get:
y ≈ 2,254 feet
Therefore, the height of the rocket at a time of 7.8 seconds is approximately 2,254 feet.
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A cylinder with diameter 3 centimeters and height 8 centimeters is filled with water. Decide which figures described here, if any, could hold all of the water from the cylinder. Explain & show your reasoning. 4.) Sphere with a radius of 2 centimeters.
Therefore, the sphere described here cannot hold all of the water from the cylinder.
What is diameter?Diameter is a straight line passing through the center of a circle or a sphere, and connecting two points on its circumference. It is the longest distance between any two points on a circle or a sphere, and it is equal to twice the radius. In other words, if you know the diameter of a circle or a sphere, you can find its radius by dividing the diameter by 2, and vice versa, if you know the radius, you can find the diameter by multiplying the radius by 2.
The volume of the cylinder can be calculated as follows:
Radius of cylinder = Diameter[tex]/2 = 3/2 = 1.5 cm[/tex]
Volume of cylinder[tex]= \pi r^2h = \pi (1.5)^2(8) = 56.55 cm^3[/tex]
To determine if a sphere with a radius of 2 cm can hold all of the water from the cylinder, we can calculate its volume as follows:
Volume of sphere [tex]= 4/3\pi r^3 = 4/3\pi (2)^3 = 33.51 cm^3[/tex]
Since the volume of the cylinder is greater than the volume of the sphere, the sphere cannot hold all of the water from the cylinder. Therefore, the sphere described here cannot hold all of the water from the cylinder.
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PLEASE HELP will give brainliest
The provided arithmetic expression's explicit and recursive formulas are as follows:
Recursive formula = [tex]a_{n+1}=a_{n-7}[/tex]
Explicit formular is Tn = 22 - 7n
Distinguish between explicit formula and recursive formula?An explicit formula can be used to determine the value of any phrase in a sequence. The natural numbers 1, 2, 3, 4, and so on make up the integers. If you have a recursive formula for the sequence and know the value of the (n-1)th term in the series, you can use it to calculate the value of the nth term. While explicit formulae provide the value of a specific term depending on the location, recursive formulas supply the value of a specific phrase based on the preceding term. This is the primary difference among recursive and explicit formulations.
Given:
[tex]a_1[/tex] = 15
[tex]a_2[/tex] = 22 = 15 - 7 = [tex]a_1[/tex] - 7
Recursive formular = [tex]a_{n+1}=a_{n-7}[/tex]
a = 15, d = -7
Fn = a + (n - 1)d = 15 + (n - 1)-7 = 15 - 7n + 7 = 22 - 7n
Explicit formular is Tn = 22 - 7n
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Graph attached below,
Someone please answer it’s due today help will be very appreciated
Answer:
y = [tex]\frac{4}{3}[/tex]x - 3
Step-by-step explanation:
Oof... I haven't done this in a while, but here you go! Slope-intercept form is y=mx+b, and in this one, the y-intercept is -3, just to get that out of the way. Now, the slope is the rise over the run, and I'm going to use the points (3, 1) and (6, 5) to measure the rise and run. The change in y, or the rise, is 4, while the change in x, or the run, is 3, so the slope is 4/3. Hope this was helpful!
Answer:
[tex]y = \dfrac{4}{3}x - 3[/tex]
Step-by-step explanation:
The slope-intercept form of a line is defined as:
[tex]y=mx+b[/tex],
where [tex]m[/tex] is the line's slope, and [tex]b[/tex] is the y-coordinate of its y-intercept.
We can see that the slope of the line is:
slope = rise / run = 4/3
(See the attached image for a model of rise and run.)
We can also see that the line touches the y-axis when y = -3, so this is the y-coordinate of the line's y-intercept.
With these pieces of information, we can craft the line's equation in slope-intercept form:
[tex]y = \dfrac{4}{3}x - 3[/tex]
URGENT!! Will give brainliest if correct :)
What is the first quartile of the data set represented by the box plot shown below?
A. 30
B. 18
C. 25
D. 45
The first quartile of the data set is 25.
What is the first quartile of the data set represented by the box plot?Box plot is a simple way of representing statistical data on a plot in which a rectangle is drawn to represent the second and third quartiles, usually with a vertical line inside to indicate the median value.
The first quartile and third quartile are the lower and upper side of the rectangle respectively.
In this case, the first quartile of the data set is 25.
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In circle QQ, the length of \overset{\LARGE\frown}{RS} = \frac{4}{3}\pi RS ⌢ = 3 4 π and m\angle RQS=120^\circ∠RQS=120 ∘ . Find the area shaded below. Express your answer as a fraction times \piπ.
We can start by finding the length of the arc \overset{\LARGE\frown}{TS} using the fact that the length of the arc \overset{\LARGE\frown}{RS} is \frac{4}{3}\pi times the radius of the circle. Since angle RQS is 120 degrees, arc \overset{\LARGE\frown}{TS} is \frac{1}{3} of the circumference of the circle:
arc \overset{\LARGE\frown}{TS} = \frac{1}{3} (2\pi R) = \frac{2}{3}\pi R
Next, we can find the length of the chord TS using the Law of Cosines:
TS^2 = TR^2 + RS^2 - 2(TR)(RS)\cos(\angle TRS)
Since \angle TRS is 120 degrees, we have:
TS^2 = R^2 + (4/3)^2R^2 - 2(R)(4/3)R(-1/2)
Simplifying this expression, we get:
TS^2 = \frac{25}{9}R^2
Taking the square root of both sides, we get:
TS = \frac{5}{3}R
Now we can find the height of the shaded region by drawing the altitude from the center of the circle to chord TS. This altitude bisects chord TS and is also perpendicular to it, so it divides TS into two segments of equal length:
Height = \frac{1}{2}(TS) = \frac{5}{6}R
Finally, we can find the area of the shaded region by subtracting the area of triangle RST from the area of sector RQS:
Area of sector RQS = (120/360)\pi R^2 = \frac{1}{3}\pi R^2
Area of triangle RST = (1/2)(RS)(height) = (1/2)(4/3)R(\frac{5}{6}R) = \frac{5}{9}R^2
Area of shaded region = Area of sector RQS - Area of triangle RST = \frac{1}{3}\pi R^2 - \frac{5}{9}R^2 = \frac{2}{9}\pi R^2
Therefore, the area shaded below is \frac{2}{9}\pi times the square of the radius R of the circle.
As per the given data, in circle QQ, the area shaded below is (1/27)π.
What is circumference?The circumference is the perimeter of a circle or ellipse in geometry. That is, the circumference would be the circle's arc length if it were opened up and straightened out to a line segment.
To find the area shaded below, we need to first find the area of sector RQS and then subtract the area of triangle RQS to get the shaded area.
The length of arc RS is given as 4/3π. Since the circumference of the circle is 2πr, where r is the radius, we have:
2πr = 4/3π
r = 2/3
So, the radius of the circle is 2/3.
Next, we can use the formula for the area of a sector, which is given as:
A = [tex](1/2)r^2\theta[/tex]
where r is the radius of the sector and θ is the central angle in radians.
In this case, θ is 120 degrees or 2/3π radians, so we have:
A(sector RQS) = [tex](1/2)(2/3)^2(2/3\pi)[/tex] = 4/27π
Now, we need to find the area of triangle RQS. To do this, we can use the formula for the area of a triangle, which is given as:
A = (1/2)bh
Where b is the base of the triangle and h is its height.
Since RQ is the base of triangle RQS and is also a radius of the circle, its length is 2/3. To find the height of the triangle, we can draw a perpendicular from point S to line QR and call the point of intersection T.
Since angle RQS is 120 degrees, angle RQT is 30 degrees (since the sum of the angles in a triangle is 180 degrees), and we have:
sin 30 = h/2/3
h = 1/3
So, the area of triangle RQS is:
A(triangle RQS) = (1/2)(2/3)(1/3) = 1/9
Finally, we can find the shaded area by subtracting the area of triangle RQS from the area of sector RQS:
A(shaded) = A(sector RQS) - A(triangle RQS) = (4/27π) - (1/9) = (1/27)π
Therefore, the area shaded below is (1/27)π.
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(PLEASE HELP) A ball is thrown into the air. The height, h(t), of the ball, in metres, after t seconds is
modelled by the equation:
h(t) = -4.9(1-1.39)² + 11
1) How high off the ground was the ball when it was thrown? (/2) For full marks,
include the units(/0.5) and you must include two decimal places (/0.5)
2) What was the maximum height of the ball? (/1) For full marks, include the
units(/0.5).
3) How high was the ball at 2.5s? (/2) For full marks, include the units(/0.5) and you
must include two decimal places (/0.5)
4) is the football in the air after 6 s? (/1)
5) When does the ball hit the ground? For full marks, include the units(/0.5) and you
must include two decimal places (/0.5)
The ball was thrown 11 m off the ground. The maximum height of the ball was 11 m. The height of the ball at 2.5s was 9.07 m. Yes, the football is in the air after 6s and the ball hits the ground at 8.67 s.
What is height?Height is the measure of vertical distance or the vertical extent of an object, person, or other thing from top to bottom. It is typically measured in units of meters or feet. Height is a measure of vertical distance or elevation above a given level, most commonly sea level. Height can also be determined by measuring the altitude of an object or location. Height is an important factor in many everyday contexts, such as architecture and sports.
To calculate the answers, we must first solve for the equation of the height of the ball h(t).
h(t) = -4.9(1-1.39)² + 11
1) The ball was thrown 11 m off the ground.
To calculate this, we set t = 0.
h(0) = -4.9(1-1.39)² + 11
h(0) = 11
Therefore, the ball was thrown 11 m off the ground.
2) The maximum height of the ball was 11 m.
To calculate this, we set the derivative of the equation, h'(t) = 0 and solve for t.
h'(t) = -9.8(1-1.39)
h'(t) = 0
1-1.39 = 0
1 = 1.39
Therefore, t = 1.
Substituting t = 1 into the equation for h(t), we get:
h(1) = -4.9(1-1.39)² + 11
h(1) = 11
Therefore, the maximum height of the ball was 11 m.
3) The height of the ball at 2.5s was 9.07 m.
To calculate this, we substitute t = 2.5 into the equation for h(t).
h(2.5) = -4.9(1-1.39)² + 11
h(2.5) = 9.07
Therefore, the height of the ball at 2.5s was 9.07 m.
4) Yes, the football is in the air after 6s.
To calculate this, we substitute t = 6 into the equation for h(t).
h(6) = -4.9(1-1.39)² + 11
+ 11
h(8.67) = 0
Therefore, the ball hits the ground at 8.67 s.
In conclusion, the ball was thrown 11 m off the ground. The maximum height of the ball was 11 m. The height of the ball at 2.5s was 9.07 m. Yes, the football is in the air after 6s and the ball hits the ground at 8.67 s.
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Lesson
6-10
Find the values of x and y,
given that AABC ~ AMNC.
B
Sy
A
Lv=600
4x/2x
C
M
60
N
We can find y by solving for the length of CN: CN = Sy - BC = 600 - 0 = 600 So the values of x and y are:
x = 0 , y = 600
First, we can look at the corresponding sides of triangle ABC and triangle MNC. Since we know that these triangles are
similar (as indicated by the ~ symbol), we can set up the following proportion:
AB/AM = BC/CN
We can substitute in the given values for these lengths:
4x/2x = (Lv + Sy)/Sy
Simplifying this equation, we can cancel out the common factor of 2x on the left side:
2 = (Lv + Sy)/Sy
Multiplying both sides by Sy, we can isolate the sum of Lv and Sy:
2Sy = Lv + Sy
Combining like terms, we get:
Sy = Lv
So we now know that Sy is equal to Lv, which is 600.
To solve for x and y, we can use the fact that corresponding sides are in proportion. Specifically, we can look at the ratio of the length of AB to the length of AM:
AB/AM = 4x/Lv
Substituting in the values we know, we get:
AB/AM = 4x/600
We can also look at the ratio of the length of BC to the length of CN:
BC/CN = 2x/Sy
Substituting in the values we know, we get:
BC/CN = 2x/600
Since AB and BC are corresponding sides in triangle ABC and AM and CN are corresponding sides in triangle MNC, we
know that these ratios must be equal:
4x/600 = 2x/600
Simplifying this equation, we get:
4x = 2x
Subtracting 2x from both sides, we get:
2x = 0
Dividing both sides by 2, we get:
x = 0
Since x is 0, we can't solve for it any further. However, we can solve for y by using the ratio we found earlier:
AB/AM = 4x/600 = 0/600 = 0
Since AB is 0, we know that BC is also 0 (since the sum of the lengths of the sides of a triangle must be greater than 0).
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A company makes wax candles in the shape of a cylinder. Each candle has a radius of 4 inches and a height of 3 inches. How much wax will the company need to make 120 candles? Use 3.14 for pie , and do not round your answer.
Answer:
The volume of a cylinder is given by the formula: V = πr^2h where r is the radius and h is the height. Substituting the given values, we get: V = π(4 in)^2(3 in) V = π(16 in^2)(3 in) V = 48π in^3 Therefore, the volume of wax needed to make one candle is 48π cubic inches. To find the amount of wax needed for 120 candles, we can multiply the volume of one candle by the number of candles: Amount of wax = 48π in^3/candle × 120 candles Amount of wax = 5760π in^3 So the company will need 5760π cubic inches of wax to make 120 candles.