To find the production level that will minimize the average cost, we need to differentiate the cost function with respect to x and set it equal to zero. So:
C'(x) = 0.005x^2 + 242x + 6000
0 = 0.005x^2 + 242x + 6000
Using the quadratic formula, we get:
x = (-242 ± sqrt(242^2 - 4(0.005)(6000))) / (2(0.005))
x = (-242 ± sqrt(146416)) / 0.01
x = (-242 ± 382) / 0.01
x = -14,000 or 27,000
Since the production level cannot be negative, we can discard the negative solution and conclude that the production level that will minimize the average cost is 27,000 units.
To find the minimal average cost, we need to plug the production level back into the cost function and divide by the production level. So:
C(27,000) = 6000 + 242(27,000) + 0.005(27,000)^2
C(27,000) = 6,594,000
Average cost = C(27,000) / 27,000
Average cost = 6,594,000 / 27,000
Average cost ≈ 244.22
Therefore, the minimal average cost is approximately $244.22.
To answer your question, first, let's correct the cost function, which should be in the form of C(x) = Fixed cost + Variable cost. Assuming it is C(x) = 6000 + 242x + 0.005x^2.
A) To find the production level that will minimize the average cost, we need to first determine the average cost function, which is AC(x) = C(x)/x. So, AC(x) = (6000 + 242x + 0.005x^2)/x.
Now, find the first derivative of AC(x) concerning x, and set it equal to zero to find the minimum point:
d(AC(x))/dx = 0
The first derivative of AC(x) is:
d(AC(x))/dx = (242 + 0.010x - 6000/x^2)
Setting this to zero and solving for x will give us the production level that minimizes the average cost:
242 + 0.010x - 6000/x^2 = 0
Now, you can solve for x using numerical methods, such as Newton-Raphson or others. After solving for x, you will get the production level that minimizes the average cost.
B) To find the minimal average cost, plug the production level x you found in part A into the average cost function, AC(x):
Minimal Average Cost = AC(production level)
This will give you the minimal average cost for the given cost function.
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Find the area inside the square and outside the circle use 3.14 for pi. please help
The area inside the square and outside the circle is 0.86 square units.
To find the area inside the square and outside the circle, we need to first find the area of the square and the area of the circle.
Let's assume that the square has sides of length 2 units, which means its area is:
Area of square = side^2 = 2^2 = 4 square units
Now, let's assume that the circle has a radius of 1 unit, which means its area is:
Area of circle = pi * radius^2 = 3.14 * 1^2 = 3.14 square units
To find the area inside the square and outside the circle, we need to subtract the area of the circle from the area of the square:
Area inside square and outside circle = Area of square - Area of circle
= 4 - 3.14
= 0.86 square units
Therefore, the area inside the square and outside the circle is 0.86 square units.
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NEED ANSWERS ASAP
Peter creates a square pyramid
model for History class. The
base of the pyramid has an
area of 20 square inches. Each
triangle has an area of 10
square inches. If Peter wants to
cover the entire pyramid with
gold paper, how much paper
will he need?
WILL GIVE BRAINLIEST TO THE CORRECT ANSWER
Peter will need 60 square inches of gold paper to cover the entire square pyramid for his History class.
You need to find the total surface area of the square pyramid that Peter creates for his History class to determine how much gold paper he will need. This can be done as follows.
1: Identify the given information.
Base area: 20 square inches
Triangle area: 10 square inches
2: Determine the number of triangles in a square pyramid.
A square pyramid has 4 triangular faces.
3: Calculate the total area of the triangular faces.
Multiply the area of one triangle by the number of triangles: 10 square inches x 4 = 40 square inches.
4: Calculate the total surface area of the pyramid.
Add the base area and the total area of the triangular faces: 20 square inches (base) + 40 square inches (triangles) = 60 square inches.
Hence, the gold paper to cover the entire square pyramid is 60 square inches.
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Maya is designing a quilt. Each piece will be in the shape of a parallelogram. She wants the base, b, of each piece to be 6 inches. The area, A, must be 42 square inches. What length should Maya use for the height, h?
The required height for which the base is 6 inches and the area of 42 square inches is 7 inches.
Each piece will be in a shape of a parallelogram so we know that the area of the parallelogram is given as
area= height * base
Now we are provided with the base of the parallelogram that is 6 inches. The area of the piece is also given as 42 sq inches. So, we can solve and get the height from the above equation.
Height (h) = area/ base
Height (h)= 42/6
Height (h)= 7 inches.
Therefore, Maya should use a height of 7 inches for each piece in order to achieve a base of 6 inches and an area of 42 sq inches.
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Solve for x. Round to the nearest tenth of a degree, if necessary.
Answer:
40.7, my answer needs to be 20+ characters sooo....
A particular base ball field is a quarter circle with a radius of 290 feet. The baseball diamond is a square with a side length of 90 feet, and bases at its vertices. What is the area of the shaded section of the field?
The area of the shaded region of the circle is A = 57,918.5 feet²
Given data ,
First, we need to find the area of the quarter circle:
Area of quarter circle = (1/4) π ( r )²
= (1/4) π ( 290 )²
= 66,018.5 feet²
Next, we need to find the area of the square:
Area of square = side²
= 90²
= 8,100 feet²
Now, we can find the area of the shaded section by subtracting the area of the square from the area of the quarter circle:
Area of shaded section = Area of quarter circle - Area of square
= 66,018.5 feet² - 8,100 feet²
= 57,918.5 feet²
Hence , the area of the shaded section of the field is 57,918.5 feet²
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Jonana has a board thats 6 ft long she wants to cut it into pieces that are each 1/4 foot long. Write an equation to represent the number of pieces she cut.
Jonana cut 24 pieces of 1/4 foot length from the 6-foot board
How to Write an equation to represent the number of pieces she cutLet "x" be the number of pieces that Jonana cut.
Each piece is 1/4 foot long.
So, the total length of all the pieces is x * 1/4 = x/4 feet.
But the total length is also 6 feet.
So we can set up the equation:
x/4 = 6
Solving for x:
x = 24
Therefore, Jonana cut 24 pieces of 1/4 foot length from the 6-foot board.
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answer questions 2-20 please 1 and 5-7 are already answered no need to correct them does not need to be correct but please have relistic answers : )
The Pythagorean Theorem with regards to the relationships between the lengths of the sides of a right triangle indicates that we get;
2. x = 51
3. x = 50
4. x = 82
8. x = 2·√(77)
9. x = √(39)
10. x = 2·√(19)
11. x = 2·√(154)
12. x = 3·√3
13. x = 6·√(13)
16. A right triangle
17. A right triangle
18. The triangle is not a right triangle
19. An obtuse triangle
20. An obtuse triangle
What is the Pythagorean Theorem?The Pythagorean Theorem states that the square of the length of the hypotenuse side of a right triangle is equivalent to the sum of the squares of the lengths of the other two sides.
2. x² = 45² + 24² = 2601
x = √(2601) = 51
3. x² = 30² + 40² = 2500
x = √(2500) = 50
x = 50
4. x² = 80² + 18² = 6724
x = √(6724) = 82
x = 82
8. According to the Pythagorean Theorem, in the right triangle we get;
x² = 18² - 4² = 308
x = 2·√(77)
9. x² = 8² - 5² = 39
x = √(39)
10. x² = 20² - 18²
x² = 76
x = √(76) = 2·√(19)
x = 2·√(19)
11. x² = 25² - 3² = 616
x = √(616) = 2·√(154)
x = 2·√(154)
12. x² = 6² - 3² = 27
x = √(27)
x = 3·√3
13. x² = 22² - 4² = 468
x = √(468) = 6·√(13)
x = 6·√(13)
16. A triangle is a right triangle if the square of the side that is the longest is equivalent to the square of the other two sides, therefore;
17² = 289
15² + 8² = 289
Therefore, the triangle is a right triangle
17. 45² = 2025
27² + 36² = 2025
Therefore, the triangle is a right triangle
18. 11² = 121
9² + 4² = 97
Therefore, the triangle is not a right triangle
19. 6² = 36
4² + 3² = 25
The square of the side that is longest is larger than the sum of the squares of the other two sides, which indicates that the angle facing the longest side is lar1ger than 90°, and the triangle is an obtuse triangle.
20. 16² = 256
9² + 11² = 202
16² > 9² + 11²; Therefore, the triangle is an obtuse triangle
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Simplify the expression five to the third power +3(5-3)
Answer:
Step-by-step explanation:
The expression +3(5-3) can be simplified using the order of operations (PEMDAS) as follows: first, we need to perform the exponentiation operation, which gives us 125.
Then, we need to perform the operation inside the parentheses, which gives us 6. Finally, we multiply 3 by 6, giving us 18. Therefore, the simplified expression is 125 + 18 = 143.
To further explain this solution, we use the order of operations, which is a set of rules that dictate the order in which operations must be performed when evaluating an expression. The acronym PEMDAS stands for Parentheses, Exponents, Multiplication and Division, and Addition and Subtraction, which represents the order of operations from left to right.
In this expression, we first need to evaluate the exponentiation operation, which is 5 to the third power. This gives us 125. Next, we need to perform the operation inside the parentheses, which is 5-3. This gives us 2. We then multiply 3 by 2, which gives us 6. Finally, we add 125 and 6, giving us 131.
It is important to follow the order of operations when simplifying an expression to ensure that we obtain the correct result. By using PEMDAS, we can systematically simplify an expression step-by-step, avoiding any potential errors and obtaining a clear and concise solution.
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Find the volume of the cone in cubic feet. Use 3. 14 for π and round your answer to the nearest tenth.
(1 m ≈ 3. 2808 ft)
135,648 ft3
3,843. 8 ft3
4,790,189. 2 ft3
444,925. 4 ft3
The volume of the cone in cubic feet is approximately 0.0 cubic feet
We need to convert the given dimensions from inches to feet before calculating the volume in cubic feet.
The height of the cone is 6 inches, which is equivalent to 6/12 = 0.5 feet (since there are 12 inches in 1 foot).
The radius of the cone is 4.5 inches, which is equivalent to 4.5/12 = 0.375 feet.
Using the formula for the volume of a cone, which is:
V = (1/3) * π * r^2 * h
Substituting the given values, we get:
V = (1/3) * 3.14 * (0.375 feet)^2 * 0.5 feet
V ≈ 0.0221 cubic feet
Rounding this to the nearest tenth gives:
V ≈ 0.0 cubic feet
Therefore, the volume of the cone in cubic feet is approximately 0.0 cubic feet. None of the given options match this result.
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For y = 126 sqrt(x), find dy, given x = 9 and Δx = dx = 0.17.
For the given function y = 126 \sqrt(x), dy is 3.57 if x = 9 and Δx = dx = 0.17.
To find the change in y (or dy), we need to use the formula:
dy = f'(x) * dx
where f'(x) is the derivative of y with respect to x.
The given function is y = 126 \sqrt(x).
To find the derivative, we can use the power rule and chain rule of differentiation.
y = 126x^{1/2}
Taking the derivative of y with respect to x:
dy/dx = 1/2 * 126x^(-1/2)
Simplifying, we get:
dy/dx = 63/(\sqrt(x))
Now, we can substitute x = 9 into this expression to get the value of the derivative at that point:
dy/dx = 63/(\sqrt(9)) = 63/3 = 21
Next, we can use the given value of dx, which is 0.17, to find the change in y or dy.
dy = f'(x) * dx ≈ 21 * 0.17 ≈ 3.57
Therefore, the change in y or dy is approximately 3.57.
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Solve systems of Equation by substitution method
3(x-2) + y = 7
4x - 3(y-1) = 16
The values of the variables are x = 4 and y = 1
How to solve the equationWe have the equations as;
3(x-2) + y = 7
4x - 3(y-1) = 16
Using the substitution method
Make 'y' the subject of formula from equation (1), we have;
y = 7 - 3(x-2)
Now, substitute the expression as y in equation (2), we get;
4x - 3(7 - 3(x-2) - 1) = 16
expand the bracket, we have;
4x - 3(7 - 3x + 6 - 1) = 16
collect the like terms
4x - 3(12 - 3x) = 16
4x - 36 + 9x = 16
collect the like terms
13x = 52
x = 4
Substitute the value
y = 7 - 3(4 -2 )
y = 7 -3(2)
y = 1
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< ABC ≈ < DEF
False
True
Answer:
True (I think)
Step-by-step explanation:
Same pattern.
A -> B -> C.
D -> E -> F.
Would be false if either one didn't share the same pattern.
Air is being pumped into a spherical balloon so that its volume increases at a rate of 100cm³/s. How fast is the radius of the balloon increasing when the diameter is 50cm?
V = 4/3 πr³
When the diameter of the balloon is 50 cm, the radius of the balloon is increasing at a rate of approximately 0.0254 cm/s
How to find the radius of the balloon increasing when the diameter is 50cm?We are given that the volume of a spherical balloon is increasing at a rate of 100 cm³/s. We need to find how fast the radius of the balloon is increasing when the diameter is 50 cm.
Let's first find the expression for the volume of the balloon in terms of its radius.
V = 4/3 πr³
Differentiating with respect to time (t), we get:
dV/dt = 4πr² (dr/dt)
We are given that dV/dt = 100 cm³/s. When the diameter of the balloon is 50 cm, the radius is 25 cm.
Substituting these values, we get:
100 = 4π(25)² (dr/dt)
Simplifying, we get:
dr/dt = 100 / (4π(25)²)
dr/dt ≈ 0.0254 cm/s
Therefore, when the diameter of the balloon is 50 cm, the radius of the balloon is increasing at a rate of approximately 0.0254 cm/s
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one serving of almonds is 1/3 cup. booker bought 2 and 2/3 cups of almonds how many servings of almonds did booker buy
Booker bought 8 servings of almonds.
How many servings of almonds did Booker buy if he purchased 2 and 2/3 cups of almonds, and one serving of almonds is 1/3 cup?
One serving of almonds is equal to 1/3 cup.
To find how many servings of almonds Booker bought, we can divide the total amount of almonds he purchased by the amount in one serving:
2 and 2/3 cups of almonds = 8/3 cups of almonds
Number of servings = (total amount of almonds purchased) / (amount in one serving)
Number of servings = (8/3) / (1/3)
Number of servings = 8/3 x 3/1
Number of servings = 8
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the diagonals of a rhombus are 8 and 10cm respectively. find the area of the rhombus
[tex]\sf Let \ d_1 \ and \ d_2 \ be \ the \ lengths \ of \ the \ sides \ of \ diagonals.[/tex]
[tex]\sf Given \ that \ d_1=8 \ cm[/tex]
[tex]\sf And \ d_2=10 \ cm[/tex]
[tex]\therefore\sf Area \ of \ rhombus=\dfrac{1}{2} (d_1)(d_2)=\dfrac{1}{2}(8)(10)=40 \ cm^2[/tex]
[tex]\rightarrow\boxed{\sf Area \ of \ rhombus=40 \ cm^2}[/tex]
Troy went to a Westwood Wasps basketball game on Saturday night. He paid $86.25 for a ticket to the game. He also had to pay for the 3 hours his car was parked in the parking garage. Troy spent a total of $99.
Which equation can you use to find the cost, x, for each hour Troy's car was parked in the garage?
The equation is 86.25 + 3x = 99.
How to determine the equation that can be used to find the cost per hour for Troy's car parking in the garage?Let's assume the cost for each hour Troy's car was parked in the garage is x dollars.
Since Troy spent a total of $99, we can set up an equation based on the given information.
The cost of the ticket to the basketball game is $86.25, and Troy also had to pay for 3 hours of parking. Therefore, the equation can be written as:
86.25 + 3x = 99
In this equation, 86.25 represents the cost of the ticket, 3x represents the cost of parking for 3 hours at a rate of x dollars per hour, and 99 represents the total amount Troy spent.
Therefore, the equation is 86.25 + 3x = 99.
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andy can run 20 miles in 6 hours. how much miles can he run in 1 hour.
I see that this is a rate of change time of problem
If andy can run 20 miles in 6 hours, than our goal is to find out how much he runs in a hour
So, we just have to divide 20 by 6.
It's sorta hard to explain
Questions?
Anyhow, the answer is around 3.3 miles per hour
Answer:
3.33 miles
Step-by-step explanation:
We Know
Andy can run 20 miles in 6 hours.
How many miles can he run in 1 hour?
We Take
20 / 6 ≈ 3.33 miles
So, Andy can run about 3.33 miles in 1 hour.
Review the equation used in writing a partial fraction decomposition.
StartFraction negative 15 x + 10 Over (5 x minus 2) squared EndFraction = StartFraction A Over 5 x minus 2 EndFraction + StartFraction B Over (5 x minus 2) squared EndFraction
Which system of equations can be used to determine the values of A and B?
The answer is B. 5A=-15 -2A+B=10
have a lovely day my darlings <3
To determine the values of A and B, we can use the following system of equations: 1. 5A = -15 2. -2A + B = 10 This system of equations can be used to find the values of A and B.
To review the equation used in writing a partial fraction decomposition, we start with a fraction that has a denominator that can be factored into linear or quadratic factors. The partial fraction decomposition separates the fraction into a sum of simpler fractions, each with a single linear or quadratic factor in the denominator.
The equation you provided for partial fraction decomposition is:
StartFraction negative 15 x + 10 Over (5 x minus 2) squared EndFraction = StartFraction A Over 5 x minus 2 EndFraction + StartFraction B Over (5 x minus 2) squared EndFraction
This equation shows that the original fraction can be decomposed into two simpler fractions, one with a linear factor of (5x - 2) in the denominator (A/(5x - 2)), and one with a quadratic factor of (5x - 2)² in the denominator (B/(5x - 2)²).
To determine the values of A and B, we need to solve for them using a system of equations. In this case, we can use the coefficients of x in the numerator of each fraction to create the following system of equations:
5A = -15
-2A + B = 10
We get the first equation by setting the numerator of the first fraction (A/(5x - 2)) equal to -15x + 10, and the second equation by setting the numerator of the second fraction (B/(5x - 2)²) equal to -15x + 10 and subtracting the first equation from it.
Solving this system of equations gives us:
A = -3
B = 5
Therefore, the partial fraction decomposition of the original fraction is:
StartFraction negative 15 x + 10 Over (5 x minus 2) squared EndFraction = StartFraction -3 Over 5 x minus 2 EndFraction + StartFraction 5 Over (5 x minus 2) squared EndFraction
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Will give brainliest
find the area of this triangle.
round to the nearest tenth.
12 cm
330
5.5 cm
[ ? ] cm2
Answer:
30 cm²
Step-by-step explanation:
Formula : [tex]Area = \frac{hb}{2}[/tex]
Write a division expression for each fraction 1/2
Answer: 1 ÷ 2
Step-by-step explanation: I think this is what you mean?
2-3 Consider the indefinite integral da. The substitution (3x - 2)2 u = 3x – 2 transforms the integral into: / None of these options are correct. 1-2 3 3 du 22 u 7 s". du 9u2 du u2 s " 0 7 u du u2
The substitution (3x - 2)2 u = 3x - 2 transforms the indefinite integral da into none of the given options. It should result in the integral of the function being expressed in terms of u rather than x.
This substitution is an example of using a change of variables to simplify an integral by transforming it into a more manageable form. This can be particularly useful when dealing with complicated integrals that are difficult to solve by other methods. Additionally, using such transforms can often provide insight into the underlying structure of the problem being studied.
Based on your question, it appears that you want to perform a substitution to transform the indefinite integral of "da" using the substitution (3x - 2)² u = 3x - 2. However, the given integral "da" doesn't seem to be correct or complete. Please provide the complete integral, and I will be happy to help you with the transformation using the given substitution.
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5. Vanessa and Nancy plan to make a birthday cake. Working together, Vanessa and Nancy can complete the
birthday cake in 2 hours. If Nancy works alone, it will take her 3 times as long as it would take Vanessa to
complete the birthday cake. The equation below represents this situation.
2 2
-+-=1
3x
How many hours would it take Nancy to complete the birthday cake if she worked alone?
X
Using an equation, if Nancy worked alone, the number of hours it would take her to complete the birthday cake is 1 hour 30 minutes.
What is an equation?An equation is a mathematical statement that proves the equality or equivalence of two or more mathematical expressions.
Equations use the equal symbol (=) unlike algebraic expressions, which combine variables with numbers, constants, and values using mathematical operands.
The number of hours for Vanessa and Nancy working together to make a birthday cake = 2 hours
The number of hours it takes Vanessa to complete the cake working alone = x
The number of hours it takes Nancy to complete the cake alone = 3x
Equation:3x + x = 2
4x = 2
x = 0.5 = 30 minutes
The total time for Nancy to complete the cake = 3x = 1.5 (3 x 0.5)
= 1 hour 30 minutes
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Identify the random variable in each distribution, and classify it as
discrete or continuous. Explain your reasoning.
1) The number of hits for the players of a baseball team.
2) The distances traveled by the tee shots in a golf
The random variable in the first situation is the number of hits for the players of a baseball team and in the second situation is the distance traveled by the tee shots in a golf game.
1) The random variable in this distribution is the number of hits for the players of a baseball team. This is a discrete random variable because hits are counted as whole numbers and cannot take on non-integer values.
2) The random variable in this distribution is the distance traveled by the tee shots in a golf game. This is a continuous random variable because the distances traveled can take on any value within a certain range, including non-integer values. The exact distance traveled by a tee shot can be measured to any degree of precision, and there are infinitely many possible distances within the range of possible outcomes. Therefore, it is a continuous random variable.
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Last year, Jane opened an investment account with . At the end of the year, the amount in the account had decreased by . How much is this decrease in dollars? How much money was in her account at the end of last year?
Decrease in amount:
Year-end amount:
1. The dollar decrease of Jane's investment account is $2,166.
2. The amount in Jane's account at the end was $5,434.
How much is this decrease in dollars?To get the decrease in dollars, we need to calculate 28.5% of $7,600.
The dollar decrease will be:
= 0.285 x $7,600
= $2,166
How much money was in her account?To find the amount, we must subtract the decrease from the initial amount. So, the end balance will be:
= $7,600 - $2,166
= $5,434
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Some lemon, lime, and cherry lollipops are placed in a bowl. Some have a
chocolate center, and some do not. Suppose one of the lollipops is chosen
randomly from all the lollipops in the bowl. According to the table below, if it
is known to be lime, what is the probability that it does not have a chocolate
center?
OA. 35%
OB. 25%
O C. 45%
O D. 55%
See picture of diagram, I need this correct please help asap
Answer:
A (35%)
Step-by-step explanation:
Just divide what you want to find with the total. Does not have chocolate = 7. Total Lime = 20 (13 + 7) 7/20 = 0.35/35%
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Of the last 20 contestants on a game show, 8 won a prize. What is the experimental probability that the next contestant will win a prize?
The probability that the next contestant will win a prize is 2/5.
What is probability?
Probability is a way of calculating how likely something is to happen. It is difficult to provide a complete prediction for many events. Using it, we can only forecast the probability, or likelihood, of an event occurring. The probability might be between 0 and 1, where 0 denotes an impossibility and 1 denotes a certainty.
Experimental probability is based on the result of an experiment that has been carried out multiples times.
probability that the next contestant will win a prize = number of contestants that won in the last games / total number of contestants in the last games
=> 8/20
To transform to the simplest form. divide both the numerator and the denominator by 4
=> 2/5
Hence the probability that the next contestant will win a prize is 2/5.
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Find the radius of the circle with equation x² + y² = 19²
r=0
Submit Answer
The radius of the circle of equation x² + y² = 19² is given as follows:
r = 19 units.
What is the equation of a circle?The equation of a circle of center [tex](x_0, y_0)[/tex] and radius r is given by:
[tex](x - x_0)^2 + (y - y_0)^2 = r^2[/tex]
The radius of a circle represents the distance between the center of the circle and a point on the circumference of the circle.
The equation of the circle in this problem is given as follows:
x² + y² = 19².
Hence the radius is obtained as follows, comparing to the general equation:
r² = 19²
r = 19 units.
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A textbook company will ship a box of textbooks to college students. The weight of the box in pounds can be determined by the function w(x) = 7. 5x + 2, where x is the number of textbooks in the box. The company requires a student to order at least 2 books but not more than 6 books. What is the range of the function for this situation? *
A. 2 ≤ x ≤ 6
B. 17 ≤ y ≤ 47
C. {17, 24. 5, 32, 39. 5, 47}
D. {2, 3, 4, 5, 6}
The range of the function for this situation is {17, 24.5, 32, 39.5, 47}, The correct option is C.
To find the range of the function for this situation, we need to evaluate the function for x = 2, 3, 4, 5, and 6, since these are the only valid inputs based on the problem statement.
w(2) = 7.5(2) + 2 = 17
w(3) = 7.5(3) + 2 = 24.5
w(4) = 7.5(4) + 2 = 32
w(5) = 7.5(5) + 2 = 39.5
w(6) = 7.5(6) + 2 = 47
Therefore, the range of the function for this situation is {17, 24.5, 32, 39.5, 47}, which is option C.
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Question 10 > Determine whether the integral is divergent or convergent. If it is convergent, evaluate it. If not, state your answer as "DNE" 4 ( 1 -dz 22 +1 00
To determine the convergence of the integral 4(1-z)/(2^2 +1)² dz from 0 to infinity, we can use the comparison test.
First, note that (1-z) is bounded between 0 and 1, so we can compare it to the integral of 4/(2² +1)² dz from 0 to infinity, which is convergent since it is a constant times the convergent p-series with p=2.
Therefore, by the comparison test, the original integral is also convergent. To evaluate it, we can use partial fractions:
4(1-z)/(2² +1)^2 = A/(2+i)² + B/(2-i)²
Solving for A and B, we get A = (1+i)/5 and B = (1-i)/5.
Then, the integral becomes:
∫ 4(1-z)/(2² +1)² dz = A ∫ 1/(2+i)² dz + B ∫ 1/(2-i)² dz
= (1+i)/5 [-1/(2+i)] + (1-i)/5 [-1/(2-i)] from 0 to infinity
= [(1+i)(2-i) - (1-i)(2+i)]/25(2² +1)
= 0
Therefore, the integral is convergent and evaluates to 0.
To determine if an integral is convergent or divergent, you need to look at its limits and the function you are integrating. Convergent means that the integral has a finite value, while divergent means the integral does not have a finite value.
For example, if you have an integral like this:
∫(f(x) dx) from a to b,
you need to evaluate the limits 'a' and 'b' and the function 'f(x)'. If 'a' or 'b' is infinity (∞) or the function 'f(x)' behaves such that it leads to an infinite value for the integral, it is divergent. Otherwise, it is convergent.
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A researcher is studying life expectancy in different parts of the world using birth and death records she randomly selects a sample of 20 people from town a and a samle of 20 people form town b and records their lifespans in years.
mean lifespan in years standard deviation
town a 78.5 11.2
town b 74.4 12.3
the researhers wants to test the claim that there is a significant differnce in lifepan for people in the two towns. what are the null and alternative hypotheses that should be used to test this claim?
a. null: mu1 - mu2 is not equal to 0; alternative: mu1 - mu2 > 0 <---- a
b. null: mu1 - mu2 = 0; alternative: p1 - p2 is not equal to 0 <---- b
c. null: mu1 - mu2 = 0; alternative: mu1 - mu2 is not equal to 0 <---- c
d. null: p1 - p2 = 0; alternative: p1 - p2 < 0 <---- d
The correct answer is: c. null: mu1 - mu2 = 0; alternative: mu1 - mu2 is not equal to 0
The null hypothesis states that there is no significant difference in the mean lifespan of people in the two towns, while the alternative hypothesis states that there is a significant difference in the mean lifespan of people in the two towns.
Since we are comparing the means of two populations, we use the difference between the means as the test statistic. Therefore, the null hypothesis is mu1 - mu2 = 0 (there is no difference between the means) and the alternative hypothesis is mu1 - mu2 is not equal to 0 (there is a difference between the means).
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