The unit vector in the direction of maximum increase of f(x,y,z) at P is:
v = (∇f(-1,-1,-1)) / ||∇f(-1,-1,-1)|| = (2/2√3, 2/2√3, 2/2√3) = (√3/3, √3/3, √3/3)
a. The gradient of f(x,y,z) is given by the vector ∇f(x,y,z) = (∂f/∂x, ∂f/∂y, ∂f/∂z). Using the partial derivative rules, we have:
∂f/∂x = 2yz
∂f/∂y = 2xz
∂f/∂z = 2xy
Therefore, the gradient of f(x,y,z) is:
∇f(x,y,z) = (2yz, 2xz, 2xy)
Evaluating this at P(-1,-1,-1), we get:
∇f(-1,-1,-1) = (2(-1)(-1), 2(-1)(-1), 2(-1)(-1)) = (2,2,2)
b. The unit vector in the direction of maximum increase of f(x,y,z) at P is given by the unit vector in the direction of ∇f(-1,-1,-1). Since ∇f(-1,-1,-1) = (2,2,2), the unit vector in the direction of ∇f(-1,-1,-1) is:
v = (∇f(-1,-1,-1)) / ||∇f(-1,-1,-1)||
where ||∇f(-1,-1,-1)|| is the magnitude of the gradient vector, which is:
||∇f(-1,-1,-1)|| = sqrt((2)^2 + (2)^2 + (2)^2) = 2√3
Therefore, the unit vector in the direction of maximum increase of f(x,y,z) at P is:
v = (∇f(-1,-1,-1)) / ||∇f(-1,-1,-1)|| = (2/2√3, 2/2√3, 2/2√3) = (√3/3, √3/3, √3/3)
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Jack starts to save at age 40 for a vacation home that he wants to buy for his 50th birthday. He will contribute $1000 each quarter to an account, which earns 2. 1% interest, compounded annually. What is the future value of this investment, rounded to the nearest dollar, when Jack is ready to purchase the vacation home?
$11,000
$11,231
$44,000
$44,924
The future value of the investment when Jack is ready to purchase the vacation home is $44,924.
To solve this problem, we can use the formula for future value of an annuity:
FV = Pmt x [(1 + r)^n - 1] / r
Where:
Pmt = $1000 (quarterly contribution)
r = 0.021 (annual interest rate)
n = 40 (number of quarters until Jack turns 50)
Plugging in the numbers, we get:
FV = $1000 x [(1 + 0.021)^40 - 1] / 0.021
FV = $44,924.38
Therefore, the future value of Jack's investment, rounded to the nearest dollar, is $44,924. So the correct answer is $44,924.
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For this problem, a table has been started for you based on the information given in the problem. use inductive reasoning to complete the table.
an electronics store finds that over a period of three months, sales of stereos decreased. in march, the store sold 325 stereos. in april, the store sold 280 stereos, and in may, the store sold 235 stereos.
month
stereos sold
march
325
april
280
may
235
june
july
august
incorrect feedback has been removed from the screen.
type your answers and then click or tap done.
make a conjecture about the number of stereos sold in june. fill in the blank text field 1
190
make a conjecture about the number of stereos sold in july.
make a conjecture about the number of stereos sold in august.
Using inductive reasoning, we can observe a pattern in the given data: the number of stereos sold decreases by 45 each month.
We can apply this pattern to make conjectures about the number of stereos sold in June, July, and August.
June: 235 (May's sales) - 45 = 190 stereos
July: 190 (June's sales) - 45 = 145 stereos
August: 145 (July's sales) - 45 = 100 stereos
So, the conjectures for the number of stereos sold are:
June: 190
July: 145
August: 100
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Carlos spots an airplane on radar that is currently approaching in a straight line, and that will fly directly overhead. the plane maintains a constant altitude of 7275 feet. carlos initially measures an angle of elevation of 20°
∘
to the plane at point aa. at some later time, he measures an angle of elevation of 37°
∘
to the plane at point bb. find the distance the plane traveled from point aa to point bb. round your answer to the nearest foot if necessary.
The distance the plane traveled from point A to point B is approximately y - x:
Distance = y - x
≈ 14046.99 feet - 20246.71 feet
≈ -6200.72 feet.
To find the distance the plane traveled from point A to point B, we can use trigonometry and the concept of similar triangles.
Let's denote the distance from point A to the plane as x, and the distance from point B to the plane as y. We are given the altitude of the plane (constant) as 7275 feet.
At point A, Carlos measures an angle of elevation of 20 degrees to the plane, and at point B, he measures an angle of elevation of 37 degrees to the plane.
Using trigonometry, we can set up the following equations:
tan(20 degrees) = 7275 / x,
tan(37 degrees) = 7275 / y.
We can rearrange these equations to solve for x and y:
x = 7275 / tan(20 degrees),
y = 7275 / tan(37 degrees).
Using a calculator, we can evaluate these expressions:
x ≈ 20246.71 feet,
y ≈ 14046.99 feet.
Therefore, the distance the plane traveled from point A to point B is approximately y - x:
Distance = y - x
≈ 14046.99 feet - 20246.71 feet
≈ -6200.72 feet.
Since the distance cannot be negative, we can round the absolute value of the result to the nearest foot:
Distance ≈ 6201 feet.
To find the distance the plane traveled from point A to point B, we can use trigonometry and the concept of similar triangles.
Let's denote the distance from point A to the plane as x, and the distance from point B to the plane as y. We are given the altitude of the plane (constant) as 7275 feet.
At point A, Carlos measures an angle of elevation of 20 degrees to the plane, and at point B, he measures an angle of elevation of 37 degrees to the plane.
Using trigonometry, we can set up the following equations:
tan(20 degrees) = 7275 / x,
tan(37 degrees) = 7275 / y.
We can rearrange these equations to solve for x and y:
x = 7275 / tan(20 degrees),
y = 7275 / tan(37 degrees).
Using a calculator, we can evaluate these expressions:
x ≈ 20246.71 feet,
y ≈ 14046.99 feet.
Therefore, the distance the plane traveled from point A to point B is approximately y - x:
Distance = y - x
≈ 14046.99 feet - 20246.71 feet
≈ -6200.72 feet.
Since the distance cannot be negative, we can round the absolute value of the result to the nearest foot:
Distance ≈ 6201 feet.
Therefore, the distance the plane traveled from point A to point B is approximately 6201 feet.
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Consider the governing equation of a system. The coefficient 'a' in the equattion is a positive constant.First, let a=4. What is the value of x in steady state? Suppose that coefficient has changed to a=2. What is the new value of x in the steady state?
To answer this question, we need to know the specific governing equation of the system. Without this information, we cannot determine the value of x in steady state for either case.
However, we do know that the coefficient 'a' in the equation is a positive constant. When a=4, we can solve for x in steady state using the given equation and the value of a=4. When a=2, we can solve for x in steady state using the same equation and the new value of a=2.
In general, the value of x in steady state will depend on the specific equation and the values of its coefficients.
Hi there! To help you with your question, I need more information about the governing equation of the system. Please provide the complete equation with 'x' and the coefficient 'a'. Once I have that information, I can help you find the steady-state values of x for a=4 and a=2.
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Calculate A. ∂z and ∂x
B. ∂z and ∂y
at the point
(5, 17, 1)
where z is defined implicitly by the equation
z4 + z2x2 − y − 9 = 0
At the point (5, 17, 1), the partial derivatives of z with respect to x and y are -12.5 and 0.25, respectively, as calculated using implicit differentiation. At the point (5, 17, 1), the partial derivatives of z with respect to z and y are 0.16 and -1.
To find the partial derivatives, we need to use the implicit differentiation.
To find ∂z/∂x, we differentiate the equation with respect to x, treating y and z as functions of x
4z^3(dz/dx) + 2z^2x^2 - 0 - 0 = 0
Simplifying, we get
4z^3(dz/dx) = -2z^2x^2
(dz/dx) = -1/2x^2z
At the point (5, 17, 1), we have
(dz/dx) = -1/2(5)^2(1) = -12.5
To find ∂z/∂y, we differentiate the equation with respect to y, treating x and z as functions of y
4z^3(dz/dy) - 1 - 0 + 0 = 0
Simplifying, we get
4z^3(dz/dy) = 1
(dz/dy) = 1/4z^3
At the point (5, 17, 1), we have
(dz/dy) = 1/4(1)^3 = 0.25
To find ∂z and ∂y at the point (5, 17, 1), we need to take partial derivatives with respect to z and y, respectively, of the implicit equation
z^4 + z^2x^2 - y - 9 = 0
Taking the partial derivative with respect to z, we get
4z^3 + 2z^2x^2(dz/dz) - dy/dz = 0
Simplifying and solving for ∂z, we get
∂z = dy/dz = 8z^3/(2z^2x^2) = 4z/x^2
At the point (5, 17, 1), we have
z = 1, x = 5
So, ∂z at the point (5, 17, 1) is
∂z = 4z/x^2 = 4(1)/(5^2) = 0.16
To find ∂y, we take the partial derivative with respect to y, keeping x and z constant
-1 = ∂y
Therefore, ∂y at the point (5, 17, 1) is -1.
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given a standard deck of cards, what is the probability of choosing a diamond, then a heart, then a black card if no replacement is made
Answer:The probability of both is 1/4*13/51.
Step-by-step explanation:
There are 52 cards in the deck, 13 hearts and 13 spades. The probability of getting a heart is 13/52 or 1/4. Given an initial heart there are 51 cards remaining; the probability of a spade is now 13/51
northview swim club has a number of members on monday. on tuesday, 22 new members joined the swim clun on wednesday 17 members cancled their membership or left the swim clun northview swim club has 33 members on thursday morning the equation m+22-17=33 repersents the situation solve the equation
There were 28 members in the Northview Swim Club on Monday before any new members joined or any current members left.
What is the solution of the equation?The equation "m+22-17=33" represents the situation where "m" is the number of members in the Northview Swim Club on Monday.
To solve the equation, we can start by simplifying it:
m + 5 = 33
Next, we can isolate "m" on one side of the equation by subtracting 5 from both sides:
m = 33 - 5
m = 28
Thus, the solution of the equation for the Northview Swim Club on Monday before any new members joined is determined as 28 members.
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Show your work for multiplying the polynomials below and put your answer in standard form in the box below: (No work loses points)
(x+6)(x2−3x−4)
The polynomials are multiplied to give the expression x³ + 3x² - 22x - 24
How to determine the productWe need to know that algebraic expressions are described as expressions that are composed of terms, variables, their coefficients, factors and constants.
Also, these expressions are made up of mathematical operations. They are listed as;
SubtractionMultiplicationDivisionAddition BracketParenthesesFrom the information given, we have the expression;
(x+6)(x2−3x−4)
expand the bracket, we get;
x³ - 3x² - 4x + 6x² - 18x - 24
add the like terms, we get;
x³ + 3x² - 22x - 24
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Pls help me find the exponent!
Answer:
1.6×10^-12..............
A recipe for banana pudding calls for 2/3 of a cup of sugar for the flour mixture and 1/4 of a cup of sugar for the meringue topping. How many cups of sugar in all is required to make the banana pudding?
Answer: To find the total amount of sugar required to make the banana pudding, we need to add the amount of sugar needed for the flour mixture to the amount of sugar needed for the meringue topping.
The recipe calls for 2/3 of a cup of sugar for the flour mixture and 1/4 of a cup of sugar for the meringue topping. To add these two fractions, we need to find a common denominator. The least common multiple of 3 and 4 is 12, so we can convert these fractions to twelfths:
2/3 = 8/12
1/4 = 3/12
Now we can add these two fractions:
8/12 + 3/12 = 11/12
So the total amount of sugar required to make the banana pudding is 11/12 of a cup.
For a certain company , the cost for producing x items is 50x + 300 and the revenue for selling x items is 90x - 0. 5x^2.
a) set up an expression for the profit from producing and selling x items. We assume that the company sells all of the items that it produces (hint: it is a quadratic polynomial)
b) find two values of x that will create a profit of $300
c) is it possible for the company to make a profit of $15,000
Answer:
Step-by-step explanation:
a) Profit = Revenue - Cost = (90x - 0.5x²) - (50x + 300)
= -0.5x² + 90x - 50x - 300
= -0.5x² + 40x - 300
b) -0.5x² + 40x - 300 = 300
-0.5x² + 40x - 600 = 0
use quadratic equation to find the roots of x (a = -0.5, b = 40, c = -600):
x = 20, 60
c) -0.5x² + 40x - 300 = 15000
-0.5x² + 40x - 15300 = 0
use quadratic equation to find the roots of x (a = -0.5, b = 40, c = -15300):
x = 40±10√290i
Not possible to make a profit of $15,000
How many 4-digit numbers have the second digit even and the fourth digit at least twice the second digit?
There are 1350 4-digit numbers that have the second digit even and the fourth digit at least twice the second digit.
To form a 4-digit number, we have 10 choices for each digit, except the first digit, which can't be 0. Hence, there are 9 choices for the first digit.
For the second digit, there are 5 even digits (0, 2, 4, 6, 8) to choose from.
For the third digit, there are 10 choices.
For the fourth digit, we can choose any of the even digits we picked for the second digit, or any of the larger odd digits 4, 6, 8.
Hence, the number of 4-digit numbers that meet the given criteria is
9 × 5 × 10 × 3 = 1350.
Therefore, there are 1,350 4-digit numbers that have the second digit even and the fourth digit at least twice the second digit.
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An oil tank is the shape of a right rectangular prism. The inside of the tank is 36. 5 cm long, 52 cm wide, and 29 cm
high. If 45 liters of oil have been removed from the tank since it was full, what is the current depth of oil left in the
tank?
The current depth of oil left in the tank is approximately 4.64 cm.
The volume of the oil tank can be found by multiplying its length, width, and height:
Volume of the oil tank = length x width x height
= 36.5 cm x 52 cm x 29 cm
= 53,854 cubic cm
If 45 liters of oil have been removed from the tank, the current volume of oil in the tank is:
Current volume of oil = Total volume of tank - Volume of oil removed
= 53,854 cubic cm - 45,000 cubic cm (1 liter = 1000 cubic cm)
= 8,854 cubic cm
Let's assume that the depth of oil left in the tank is x cm. Then the volume of oil left in the tank can be found by multiplying the length, width, and depth of oil:
Volume of oil left in tank = length x width x depth of oil
= 36.5 cm x 52 cm x x cm
= 1906x cubic cm
Now we can set up an equation to find the value of x:
1906x = 8,854
Dividing both sides by 1906, we get:
x = 4.64 cm
Therefore, the current depth of oil left in the tank is approximately 4.64 cm.
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Please help this is for a test and i need a good grade lollll
"the wind force f on a sail varies jointly as the area al of the sall and the square of the wind speed w.
the force on a sail with area an area of 500 p? is 64.8 pounds when the wind speed is 18 mph. what
would be the force for a sail with an area of 250 f12 with a wind speed of 35 mph"
please show step by step work tysmmmm <3
The force on a sail with an area of 250 f12 and a wind speed of 35 mph would be 108.72 pounds.
How to find force on sail?We are given that the wind force F on a sail varies jointly as the area A and the square of the wind speed W. We can represent this relationship mathematically using the equation:
F = k * A * W²
where k is a constant of proportionality.
We are also given that the force on a sail with an area of 500 p and wind speed of 18 mph is 64.8 pounds. We can use this information to solve for k:
64.8 = k * 500 * 18²
Solving for k, we get:
k = 64.8 / (500 * 18²)
k = 0.0000768
Now, we can use the equation to find the force for a sail with an area of 250 f12 and a wind speed of 35 mph:
F = 0.0000768 * 250 f12 * 35²
F = 108.72 pounds
Therefore, the force on a sail with an area of 250 f12 and a wind speed of 35 mph would be 108.72 pounds.
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Find the value of x such that the data set has the given mean.
102, 120, 103, 112, 110, x; mean 108
The value of x in the data set is 101.
How to find mean?The mean of a data set is the sum of all the data divided by the count n.
Therefore, let's find the mean of the data set as follows:
The mean is the sum of the data divided by the total number of data.
Hence, let's find the value of x using the mean
108 = 102 + 120 + 103 + 112 + 110 + x / 6
108 = 547 + x / 6
Cross multiply
108 × 6 = 547 + x
648 = 547 + x
x = 648 - 547
x = 101
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A circle is circumscribed around a regular octagon with side lemgths of 10 feet. Another circle is inscribed inside the octagon. Find the area. Of the ring created by the two circles. Round the respective radii of the circles to two decimals before calculating the area
The area of the ring is 1,462.81 square feet, under the condition that a circle is circumscribed around a regular octagon with side lengths of 10 feet.
The area of the ring formed by the two circles can be evaluated using the formula for the area of a ring which is
Area of ring = π(R² - r²)
Here
R = radius of the larger circle
r = smaller circle radius
The radius of the larger circle is equal to half the diagonal of the octagon which is 10 feet. Applying Pythagoras theorem, we can evaluate that the length of one side of the octagon is 10/√2 feet.
Radius of the larger circle is
R = 5(10/√2)
= 25√2/2 feet
≈ 17.68 feet
Staging these values into the formula for the area of a ring,
Area of ring = π(17.68² - 10²) square feet
Area of ring ≈ 1,462.81 square feet
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Help me please I don’t know what to do
Answer:
179.3 square units
Step-by-step explanation:
We have to find the area of the rectangle and area of semicircle using the formula and then add the areas.
Area of rectangle:
length = 14 units
width = 10 units
[tex]\sf \boxed{\text{\bf Area of rectangle = length * width}}[/tex]
= 14 * 10
= 140 square units
Area of semicircle:
diameter of semicircle = width of the rectangle
d = 10 units
r = d ÷ 2
= 10 ÷ 2
= 5 units
[tex]\boxed{\text{\bf Area of semicircle = $\dfrac{1}{2}\pi r^2$}}[/tex]
[tex]\sf = \dfrac{1}{2}*3.14*5*5\\\\ = 39.26\\\\ = 39.3 \ square \ units[/tex]
Area of the figure = area of rectangle + area of semicircle
= 140 + 39.3
= 179.3 square units
A 40 -degree angle is translated 5 inches along a vector. What is the angle measurement, in degrees, of the image?
The angle measurement would remain as 40 degrees
Does angle change when translated?No, when a geometric figure, such as a line or an angle, is translated (moved) to a new position without being rotated, reflected, or scaled, its shape and size do not change, and therefore its angle measure remains the same.
This property is a fundamental concept in geometry and is known as the "invariance of angle measure under translation". It means that if two angles are congruent (have the same measure) in their original position, they will remain congruent after being translated to a new position.
Hence The angle measurement would remain as 40 degrees
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Bill is walking up the steps in the Washington Monument at a rate of 30 feet per minute and Joe is walking down at the rate of 45 feet per minute. Bill is 75 feet from the bottom at the same moment that Joe is 325 feet from the bottom. Which of the following systems of equations can be used to determine the number of minutes t, from now and height, ℎ (in feet), at which they will pass each other?
The equation that can be used to determine the number of minutes t, from now and height, ℎ (in feet), at which they will pass each other is 75t = h.
What is the time taken for them to pass each other?The time taken for them to pass each other is calculated as follows;
Apply the rules of relative velocity;
(V₂ - V₁)t = h
where;
V₂ is the velocity of the BillV₁ is the velocity of the Joet is the time taken for them to meeth is the distance between them(30 ft/min - ( -45 ft/min )t = h
75t = h
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A fountain is in the shape of a right triangle. The area of the fountain is
12 square meters. One leg of the triangle measures one and a half times the
length of the other leg. What are the lengths of all three sides of the fountain?
Answer:
4,6,[tex]\sqrt{52} \\[/tex]
Step-by-step explanation:
Area of right triangle= base x height/2=12, but if we remove the division then it's:
base x height=24
factors of 24= 6,4 8,3 24,1 and 12,2
we have the rule that "One leg of the triangle measures one and a half times the length of the other leg." and the pair that matches that is 6 and 4.
So leg a=4 and leg b=6. Using the Pythagorean theorem(a^2+b^2=c^2) we have:
4^2+6^2=c^2=16+36=52 so the answer is 4,6,[tex]\sqrt{52} \\[/tex]
Breck has 22 dimes and nickels. The total value of the coins is $1. 45. How many dimes and how many nickels does Breck have?
Jenelle draws one from a standard deck of 52 cards. Determine the probability of drawing either a two or a ten? Write your answer as a reduced fraction. Answer= Determine the probability of drawing either a two or a club? Write your answer as a reduced fraction. Answer=
The probability of drawing either a two or a ten is (4+4)/52, which simplifies to 2/13.
The probability of drawing either a two or a club is (3+13)/52, which simplifies to 4/13.
For the first question: In a standard deck of 52 cards, there are four 2s and four 10s. The probability of drawing either a two or a ten is the number of successful outcomes (drawing a 2 or a 10) divided by the total number of possible outcomes (52 cards). So, the probability is (4+4)/52 = 8/52. This can be reduced to the fraction 2/13.
For the second question: There are four 2s and thirteen clubs in a standard deck of 52 cards. Since one of the 2s is a club, there are three additional 2s that are not clubs. The probability of drawing either a two or a club is the number of successful outcomes (3 additional 2s + 13 clubs) divided by the total number of possible outcomes (52 cards). So, the probability is (3+13)/52 = 16/52. This can be reduced to the fraction 4/13.
Therefore,
1) Probability of drawing either a two or a ten: 2/13
2) Probability of drawing either a two or a club: 4/13
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Chris wants to order DVD's over the internet. Each DVD costs $15. 99 and shipping the entire order costs $9. 99. If he can spend no more than $100, how many DVD's could he buy?
Since Chris can only buy whole DVDs, he can purchase a maximum of 5 DVDs within his $100 budget.
Each DVD costs $15.99, and the shipping for the entire order is $9.99.
We can use the following inequality to represent Chris's budget constraint:
15.99x + 9.99 ≤ 100
Here, x represents the number of DVDs he can buy.
To find the maximum value of x, we can rearrange the inequality:
x ≤ (100 - 9.99) / 15.99 x ≤ 90.01 / 15.99 x ≤ 5.63
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A scale drawing of a famous statue uses a scale factor of 240:1. If the height of the drawing is 1.2 feet, what is the actual height of the statue?
288 feet
241.2 feet
238.8 feet
200 feet
The height of the statue is 288 feet.
The scale factor is 240:1
Or, the ratio of the height of the statue to the height of the drawing = 240:1.
This means, for 1 unit height of drawing, the height of the statue = 240 units
Or, for 1 feet height of the drawing, the height of the statue = 240 feet.
Let us suppose the actual height of the statue to be x.
The height of the drawing = 1.2 feet (given)
So, the ratio of the height of the statue to the height of the drawing = x/1.2
But, the scale factor = 240:1 = 240/1
∴ 240/1=x/1.2
⇒x=240×1.2
⇒x=288
Hence, the height of the statue is 288 feet.
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If there are 30 people in a classroom, what is the probability that at least two have the same birthday
The probability that at least two people in a group of 30 have the same birthday is about 0.7063 or 70.63%.
To calculate the probability that at least two people in a group of 30 have the same birthday, we can use the complement rule:
P(at least 2 people have the same birthday) = 1 - P(all people have different birthdays)
The probability that the first person has a unique birthday is 1 (since there are no other people to share with yet).
The probability that the second person also has a unique birthday is 364/365 (since there are now 364 days left out of 365 that they could have a different birthday from the first person).
Similarly, the probability that the third person has a unique birthday is 363/365, and so on. So, we can write:
P(all people have different birthdays) = 1 x 364/365 x 363/365 x ... x 336/365
Using a calculator or computer program, we can evaluate this expression to be approximately 0.2937.
Therefore,
P(at least 2 people have the same birthday) = 1 - 0.2937 = 0.7063
So the probability that at least two people in a group of 30 have the same birthday is about 0.7063 or 70.63%.
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CAN SOMEONE HELP ME PLEASEEEEEEEEEEEEEE I NEED HELP :(
Answer:
for the first three, divide the number by 2
for the second three, multiply by 2
9 and 11. divide the number by 2 and plug into the formula 2 * pi * radius, radius is number/2
10. plug 7 into formula 2 * pi * radius, radius = 7
Step-by-step explanation:
radius is half the length of the circle, diameter is the full length, circumference is 2 * pi * radius
Integrate fover the given curve. f(x,y) = x+ y, C: x^2 + y^2 = 4 in the first quadrant from
(2,0) to (0,2)
The integral of f(x, y) = x + y over the given curve is 8.
To integrate the function f(x, y) = x + y over the curve C: x² + y² = 4 in the first quadrant from (2, 0) to (0, 2), we will use the line integral. Since the curve is a circle, we can parameterize it using polar coordinates as follows:
x = 2cos(θ)
y = 2sin(θ)
Now, let's find the derivatives:
dx/dθ = -2sin(θ)
dy/dθ = 2cos(θ)
Next, we substitute x and y in f(x, y):
f(x, y) = 2cos(θ) + 2sin(θ)
Now, we can set up the line integral:
∫[f(x, y) * ||dr/dθ||]dθ
Since ||dr/dθ|| = sqrt((-2sin(θ))^2 + (2cos(θ))^2) = 2, the line integral becomes:
∫[2cos(θ) + 2sin(θ)] * 2 dθ
To find the limits of integration, we can use the points (2, 0) and (0, 2). In polar coordinates, these points correspond to θ = 0 and θ = π/2.
So, the line integral becomes:
∫[4cos(θ) + 4sin(θ)]dθ from 0 to π/2
Now, we can integrate and evaluate:
[4sin(θ) - 4cos(θ)] from 0 to π/2 = [4(1) - 4(0)] - [4(0) - 4(1)] = 8
Thus, the integral of f(x, y) = x + y over the given curve is 8.
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What is the image of (5,−4) after a dilation by a scale factor of 4 centered at the origin?
The image of (5,−4) after a dilation by a scale factor of 4 centered at the origin is (20,−16)
What is the image after a dilation centered at the origin?From the question, we have the following parameters that can be used in our computation:
Point = (5,−4)
Scale factor of 4 centered at the origin
The image after a dilation centered at the origin is
Image = Point * Scale factor
Substitute the known values in the above equation, so, we have the following representation
image = (5,−4) * 4
Evaluate
image = (20,−16)
Hence, the image after a dilation centered at the origin is (20,−16)
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HELP MARKING BRAINLEIST IF CORRECT
Answer:
21.5
Step-by-step explanation:
First we can solve for c using the pythagoreom theorem. (probably didn't spell that right)
A squared + B squared = C squared
9 squared + 3 squared = c squared
81+9= c squared
90=c squared
90 square root is (rounded to the nearest tenth) 9.5
c=9.5
Then we can add 9.5+9+3= 21.5
Al has a cylindrical storage container 30 centimeters tall with a diameter of 22 centimeters. How much bird food in cubic centimeters will fit in the container? Use the formula V = Bh and approximate π using 3.14. Round your answer to the nearest tenth.
The amount of bird food in cubic centimeters will fit in the container is
11, 398. 2 cubic centimeters
How to determine the volumeThe formula that is used for calculating the volume of a cylinder is expressed with the equation;
V = π(d/2)²h
Such that the parameters of the given equation are;
V is the volume of the cylinder.d is the diameter of the cylinderh is the height of the cylinderNow, substitute the values into the formula, we have;
Volume = 3.14 (22/2)² 30
divide the values
Volume = 3.14(121)30
Now, multiply the values and expand the bracket
Volume = 11, 398. 2 cubic centimeters
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