The triangle ABC is shown in the figure below. All 3 sides of the triangle are labeled a, b, and c, and the right angle is located at the vertex C.
To verify that the triangle ABC is a right triangle, we can use the Pythagorean theorem. The Pythagorean theorem states that the sum of the squares of the two shorter sides of the right triangle is equal to the square of the longest side (hypotenuse). Therefore, in the case of triangle ABC, we can calculate a^2 + b^2 = c^2. When we calculate, we can see that the equation is true, meaning that the triangle ABC is indeed a right triangle.
To find the area of triangle ABC, we can use Heron's formula. Heron's formula states that the area of a triangle is equal to the square root of s(s-a)(s-b)(s-c), where s is the semiperimeter (the sum of the three sides divided by 2). Therefore, in the case of triangle ABC, the area is equal to the square root of s(s-a)(s-b)(s-c), which is equal to 6.
Therefore, triangle ABC is a right triangle with an area of 6.
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A construction company purchases a crane for $450,000. An accountant for the company figures the crane loses 30% of its value every year. The value of the crane can be computed after x years using the function V ( x ) = 450 , 000 ( 1 − 0.3 ) x . Use the graph to answer the following questions What is the value after 5 years? cells When will the value decline below $140000? (round to the nearest tenth).
The value of the company after 5 years is given as follows:
$75,631.5.
The value will decline below $140,000 when:
3.27 years.
What is the exponential function?The exponential function for this problem is defined as follows:
V(x) = 450000(0.7)^x.
The value after 5 years is found replacing the lone instance of x by 5, hence:
V(5) = 450000(0.7)^5
V(5) = $75,631.5.
The value will decline below $140,000 when at x for which V(x) = 140000, hence:
140000 = 450000(0.7)^x.
(0.7)^x = 140000/450000
(0.7)^x = 0.311111
xlog(0.7) = log(0.311111)
x = log(0.311111)/log(0.7)
x = 3.27 years.
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4
10 points
(x-3)²
O x² - 6x +9
O x²-9
Ox+9
O x² +9
The correct expression is a) [tex]x^2-6x+9[/tex].
What is expression?
Mathematical expressions consist of at least two numbers or variables, at least one arithmetic operation, and a statement. It's possible to multiply, divide, add, or subtract with this mathematical operation. Unknown variables, integers, and arithmetic operators are the components of an algebraic expression. There are no symbols for equality or inequality in it.
Here the given expression is ,
=> [tex](x-3)^2[/tex]
Now using formula [tex](a-b)^2=a^2-2ab+b^2[/tex] then
=> [tex]x^2-2.x.3+3^2[/tex]
=> [tex]x^2-6x+9[/tex]
Hence the correct answer is a) [tex]x^2-6x+9[/tex].
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What are the solutions to 4lx+8l=4
The two solutions that satisfy the equation 4lx+8l=4 are -9 and -7.
What is the solution of an equation?A solution is the assignment of values to the unknown variables that results in the equation's equality being true. In other words, a solution is a value or a set of values (one for each unknown) that, when substituted for the unknowns, solves the equation.To solve an equation, use the properties of real numbers as those that apply to variables to manipulate the equation.Given the equation,
4lx+8l=4
|x+8| = 4/4
|x+8| = 1
x + 8 = ± 1
When x = + 1
Then,
x = 1 - 8
x = -7
When x = - 1
x = -1 - 8
x = -9
Therefore the two solution of equation is -9 and -7.
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If you dissolved 50 grams of sugar in 30 grams of water. What would the sugar concentration be?
According to the information, the concentration of sugar in this solution is 62.5%.
How to calculate the concentration of sugar?To calculate the concentration of sugar we have to divide the mass of the solute by the mass of the solution and then multiply it for 100.
Equation:
(Mass of solute / Mass of the solution ) x 100Mass of the solute (sugar) = 50gMass of the solvent (water) = 30gMass of the solution (sugar + water) = 30g + 50g = 80gThe sugar concentration = 50g/80g x 100 = 62.5%
Therefore, the sugar concentration is 62.5%.
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Toastmasters International cites a report by Gallop Poll that 40% of Americans fear public speaking. A student believes that less than 40% of students at her school fear public speaking. She randomly surveys 361 schoolmates and finds that 135 report they fear public speaking. Conduct a hypothesis test to determine if the percent at her school is less than 40%.
Confidence Interval: (0.3241, 0.4240)
There is not enough evidence to conclude that the percent at her school is less than 40%.
How to reach the conclusion of the hypothesis tested?The conclusion of the hypothesis test is reached depending on the confidence interval as follows:
Upper bound below 40%: enough evidence to conclude that the percent at her school is less than 40%.Upper bound above 40%: not enough evidence to conclude that the percent at her school is less than 40%.The upper bound of the interval is given as follows:
0.4240 = 42.40%.
Hence not enough evidence to conclude that the percent at her school is less than 40%.
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James has $2000 in one savings account. Let m represent the amount he has in another savings account. Write an expression for the total amount of money in
both accounts.
An algebraic expression for the total amount of money in both accounts is (blank)
Answer:
An algebraic expression for the total amount of money in both accounts is m+2000
Step-by-step explanation:
If there are 2000 dollars in one account and m dollars in the other, the sum of both accounts can be expressed as 2000+m (or m+2000).
Dr. Leona Williams, a well-known plastic surgeon, has a reputation for being one of the best surgeons for reconstructive nose surgery. Dr. Williams enjoys a rather substantial degree of market power in this market. She has estimated demand for her work to be: Q = 480 – 0.2P where Q is the number of nose operations performed monthly and P is the price of a nose operation. What is the inverse demand function for Dr. Williams’ services? What is the marginal revenue function? The average variable cost function for reconstructive nose surgery is estimated to be: AVC = 2Q2 – 15Q + 400 where AVC is the average variable cost (measured in dollars), and Q is the number of operations per month. The doctor’s fixed cost per month is $8,000. If the doctor wishes to maximize her profit, how many nose operations should she perform each month? What price should Dr. Williams charge to perform a nose operation? How much profit does she earn each month?
The inverse demand function is P = 2400 + 5Q
The Marginal Revenue function is MR = 5QThe number of nose operations is 15The price per month is $2475Dr. Williams earns $19750 in profit each monthHow to determine the inverse demand functionGiven that
Q = 480 - 0.2P
We simply make P the subject of the formula
So, we have
0.2P = 480 - Q
Divide by 0.2
P = 2400 + 5Q
The marginal revenue functionTo find the marginal revenue function, we can use the inverse demand function P = 2400 + 5Q, and take the derivative of this function with respect to Q.
The derivative of P with respect to Q is dP/dQ = 5. This represents the change in price for a one unit change in the quantity of operations performed.
So the Marginal Revenue function is MR = dP/dQ * Q = 5Q
The number of nose operationsTo maximize profit, the doctor should equate marginal revenue to marginal cost, which is
MC = dAVC/dQ
So, we have
MC = d(2Q² - 15Q + 400)/dQ
Evaluate
MC = 4Q - 15.
So, we have
5Q = 4Q - 15
Q = -15
Take the absolute value
Q = 15
So, the number of operations is 15
The price for each operationWe have
P = 2400 + 5Q
This gives
P = 2400 + 5 * 15
Evaluate
P = 2475
The monthly profitTo find the profit, we need to subtract the total cost from the total revenue.
The total revenue is found by multiplying the price by the quantity of operations performed: TR = P * Q = 2475 * 15 = 37125
The total variable cost is found by multiplying the average variable cost by the quantity of operations performed: TVC = AVC * Q = (2Q^2 - 15Q + 400) * Q = 2Q^3 - 15Q^2 + 400Q
The total fixed cost is the fixed cost that doesn't change regardless of the level of output : TFC = $8000
The total cost is the sum of the total variable cost and total fixed cost: TC = TVC + TFC = 2Q^3 - 15Q^2 + 400Q + $8000
The profit is the difference between total revenue and total cost:
π = TR - TC = 37125 - (2Q^3 - 15Q^2 + 400Q + $8000)
When Q = 15,
profit is π = 37125 - (2(15)^3 - 15(15)^2 + 400(15) + $8000)
= 19750
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The inventory of Royal Decking consisted of five products. Information about the December 31, 2021, inventory is as follows:
Per Unit
Product Cost Selling Price
A $ 130 $ 150
B 170 190
C 130 170
D 90 120
E 50 70
Costs to sell consist of a sales commission equal to 10% of selling price and shipping costs equal to 10% of cost.
Required:
What unit value should Royal Decking use for each of its products when applying the lower of cost or net realizable value (LCNRV) rule
The unit value should Royal Decking use for each of its products,
Product A: $8 (loss),
Product B: $16 (loss),
Product C: $10 (profit),
Product D: $9 (profit),
Product E: $8 (profit).
What is the unit value?The sum of the quantities divided by the total value of purchases and sales yields the unit value of a group of similar goods. As a result, it represents a quantity-weighted average of the various prices at which the good is bought and sold.
Given inventory,
Product Cost Selling Price
A $130 $150
B $170 $190
C $130 $170
D $90 $120
E $50 $70
a sales commission equal to 10% of the selling price and shipping costs equal to 10%,
Product A :
the unit value is given by,
Value = selling price - (product cost + sales commission + shipping cost)
= $150 - ($130 + 10% x selling price + 10% x product cost)
= $150 - ($130 + 10% x $150 + 10% x $130)
= $150 - ($130 + $15 + $13)
= -$8 = $8 (loss)
Product B :
Value = selling price - (product cost + sales commission + shipping cost)
= $190 - ($170 + 10% x selling price + 10% x product cost)
= $190 - ($170 + 10% x $190 + 10% x $170)
= $190 - ($170 + $19 + $17)
= -$16 = $16 (loss)
Product C :
Value = selling price - (product cost + sales commission + shipping cost)
= $170 - ($130 + 10% x selling price + 10% x product cost)
= $170 - ($130 + 10% x $170 + 10% x $130)
= $170 - ($130 + $17 + $13)
= $10 = $10 (profit)
Product D :
Value = selling price - (product cost + sales commission + shipping cost)
= $120 - ($90 + 10% x selling price + 10% x product cost)
= $120 - ($90 + 10% x $120 + 10% x $90)
= $120 - ($90 + $12 + $9)
= $9 = $9 (profit)
Product E :
Value = selling price - (product cost + sales commission + shipping cost)
= $70 - ($50 + 10% x selling price + 10% x product cost)
= $70 - ($50 + 10% x $70 + 10% x $50)
= $70 - ($50 + $7 + $5)
= $8 = $8 (profit)
Hence unit value for each product is -$8, -$16, $10, $9, and $8.
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Triangular base with base edge 12 inches and base height 9 inches, and volume 108 cubic inches.
This matches the given volume of 108 cubic inches.
What is volume?Volume is a measure of the amount of space an object or substance occupies. It is measured in units such as liters, gallons, cubic meters, and milliliters. It is often used to measure the size of a container, the amount of a material, or the size of a space. Volume is an important property for understanding physical objects, as it helps to determine mass and density.
The volume of a triangular prism can be calculated by taking the area of the base, which is the area of an equilateral triangle (all sides equal) multiplied by the height of the prism. In this case, the area of the triangle is ((√3/4)*(12^2)) = 108 square inches and the height is 9 inches, so the volume is 108*9 = 108 cubic inches. This matches the given volume of 108 cubic inches.
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Work out 5 × 1 7 Give your answer as a fraction.
Answer:
The answer is 85/1. To calculate 5 × 17, first multiply 5 × 10 = 50, then add 5 to the result to get 50 + 5 = 55. Finally, divide the result by 1 to get 55/1 = 85/1.
give me brainiest
Answer:
The fraction is 5/7
Step-by-step explanation:
5 * 1/7
Multiplying a whole number times a fraction
Multiply the whole number times the numerator.
5 * 1 = 5
Divide by the denominator
5/7
The fraction is 5/7
(strang 5.2.15) consider the determinants of the n-by-n tridiagonal matrices containing only 1s for the nonzero elements:
The determinant of an n-by-n tridiagonal matrix containing only 1s can be calculated by iterating over the diagonal elements and summing the products of each element and its corresponding power of 1
The determinant of an n-by-n tridiagonal matrix containing only 1s as nonzero elements is given by the formula:
det(A) = 1^n-2 + 1^n-3 + ... + 1^2 + 1
For example, consider the 3x3 matrix:
A = [1 1 0; 1 1 1; 0 1 1]
Using the formula above, the determinant of A can be calculated as:
det(A) = 1^2 + 1 = 2
More generally, the determinant of an n-by-n tridiagonal matrix containing only 1s can be calculated by iterating over the diagonal elements and summing the products of each element and its corresponding power of 1. This sum can then be taken to the power of 1 to obtain the determinant of the matrix.
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problems 13-16, write < or > to complete each statement
Therefore , the solution of the given problem of expression comes out to be √15 -4 < √15 -7 and so on.
Expression : what is it?Multiplying, dividing, adding, and subtracting are necessary math operations. When combined, the following expression would appear: A mathematical operator, some data, and an equation Values, variables, and functions make up a statement of fact's constituent parts, such as additions, deductions, multiplications, divisions, etc. It is possible to contrast and compare words and phrases.
Here,
Given :
to find which is greatest expression among them
=> √15 -4 < √15 -7
=> 2√18 < 2√21
=>-³√30 > -3
=>√8 + 1 < √17 -2
Therefore , the solution of the given problem of expression comes out to be √15 -4 < √15 -7 and so on.
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Joey makes $9 an hour at Burger Palace. He worked 5 hours on Friday
and 8 hours on Saturday. Write an expression and use the distributive
property to find how much Joey made.
Answer:
9(5 + 8)
Step-by-step explanation:
Find the volume of the solid whose base is bounded by the circle
x2 + y2 = 4
the cross sections perpendicular to the x-axis are squares.
The volume of the solid is 32/3.
The volume of the solid whose base is bounded by the circle x2 + y2 = 4 and the cross sections perpendicular to the x-axis are squares can be calculated using the formula for the volume of a solid of revolution. This formula states that the volume is equal to the integral of the area of the cross sections from the lower limit to the upper limit.
Since the cross sections are squares, the area of each cross section is given by the equation A = x2. The lower limit is given by the equation x2 + y2 = 4, which is x=-2, and the upper limit is x=2. Thus, the volume is given by the integral ∫(-2)2x2dx = 8∫x2dx = 8(x3/3)|(-2)2 = 8(2^3/3 - (-2)^3/3) = 8(8/3 - -8/3) = 32/3. Therefore, the volume of the solid is 32/3.
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Two numbers have the sum of 19. If one of the numbers is c, express the other number in terms of c. The other number is
Two numbers have the sum of 19. If one of the numbers is c, express the other number in terms of c. The other number is 19 - c.
Given: two numbers have the sum of 19
Let: c be one of those numbers
We want to express the other number in terms of c
The other number = 19 - c
Given two numbers, we want to express one of them in terms of the other. We can let one of the numbers be represented by the variable c. Since the sum of the two numbers is 19, the other number can be expressed as 19 minus c. Therefore, the other number can be written as 19 - c.
We are given two numbers and we want to express one of them in terms of the other. We can represent one of the numbers with the variable c. Since the sum of the two numbers is 19, the other number can be expressed as 19 minus c. In other words, the other number can be written as 19 - c. This means that if the value of c is known, the value of the other number can be calculated by subtracting c from 19. For example, if c is 10, then the other number is 19 - 10 = 9. Similarly, if c is 5, then the other number is 19 - 5 = 14. Therefore, the other number can be expressed as 19 - c.
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Use the number line to find the missing value. +(-4)=-1
The value of the missing number will be 3.
What is a number line?A number line in elementary mathematics is a representation of a graduated straight line that serves as an abstraction for real numbers, represented by the symbol R."
It is assumed that every point on a number line corresponds to a real number and that every real number corresponds to a point.
The equation for the missing number is given as x + (-4) = -1. the missing number will be calculated as:-
x + (-4) = -1
x -4 = -1
x = 4 - 1
x = 3
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Part A
A rectangular prism measures 7 1/2cm by 12 cm by 15 1/2cm.
Calculate the number of cubes with edge length 1/2 cm that fit in this prism
cubes____
Answer:11160
i calculated
Given f(x) = 3x +5
find f(3) =
Answer:
f(3) = 14
Step-by-step explanation:
f(x) = 3x +5
Let x = 3
f(3) = 3*3 +5
= 9 + 5
= 14
Answer:
Step-by-step explanation:
fș
Which equation has exactly two real and two nonreal solutions?
We got two real solutions (-3,+3) and two non real solutions (-2i,+2i)
Therefore, x³+2x² +x-7=0 is correct answer.
What is real and nonreal solution?An a non real solution uses non real numbers to solve the equation, which cannot be solved with real numbers. Back to the school answer: A non-real solution to a math equation is a solution that has “imaginary numbers” (also called “complex numbers”) in it because it cannot be solved with real numbers.
Step1
We need to find equation that has exactly two real and two non real solutions
2 real and 2 non real solution means 4 solutions
x³ equation has maximum of 3 solutions
So we ignore equation that has largest exponent x³
We ignore options B and D
Let check option A
x⁴ 36x² =0
factor the left hand side
Factor out x^2
x² (x² -36)=0
WE set each factor =0 and solve for x
x² =0 so x=0
x² - 36 =0 so x= +-6
So solutions are x=0, x=6 , x= -6. Only 3 real solutions we got
LETS check with option C
x⁴ - 5x² -36=0
Factor left hand side
(x² -9)(x² +4)=0
set each factor =0 and solve for x
x² -9 =0 so x= -3, + 3
x² + 4 =0 , x² = -4, so x= +2i, -2i
So we got two real solutions (-3,+3) and two non real solutions (-2i,+2i)
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the following equation has two different solutions. solve the equation by factoring and place one solution in each answer box. the larger answer goes in box 1. the smaller solution should go in box2. solve for x: x^2 +16x + 63 = 0Box1 (Larger Solution): Box2 (Smaller Solution):
Box1: x = -9 Box2: x = 7
For this equation, we need to factor the left side of the equation to solve it. To do this, we need to find two numbers that multiply to give us 63 and add to give us 16.
The two numbers are -9 and +7.
So, the factored equation is (x - 9)(x + 7) = 0.
For this equation, we need to factor the left side of the equation to solve it. To do this, we need to find two numbers that multiply to give us 63 and add to give us 16.
To solve the equation, we set each factor equal to zero and solve for x.
x - 9 = 0 -> x = 9
x + 7 = 0 -> x = -7
Therefore, the larger solution is x = -9, and the smaller solution is x = 7.
Therefore, the answer is Box1: x = -9 and Box2: x = 7.
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Before leaving the grocery store, Kenny wrote the y-coordinate first and the x-coordinate second when telling his sister where to meet him. To his surprise, she was at the right location. If his sister did not make an error, where did the siblings meet ?- Explain how you know.
If the both coordinates are equal and has same sign then interchanging them will not have any effect and they will meet at the right location
How to find where they will meetInterchanging the x coordinate for the y coordinate when both are equal and have same sign will create no effect and hence this will not affect their meeting place
In a situation where the coordinates are different, interchanging the two coordinates will lead to meeting at a different point.
The point of meeting when the coordinates are different either in sign or in value is at reflection over the line y = x
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Consider the problem: A shopper buys cat food in bags of 3 lbs. Her cat eats 3_4lb each week. How many weeks does one bag last?
Using division operation, the number of weeks a bag of cat food lasts is 4 weeks.
What is a division operation?Division operation is one of the four basic mathematical operations.
The division operation produces a result known as the quotient.
The quotient results when the dividend is divided by the divisor.
The quantity in a bag of cat food = 3lbs
The quantity that the cat eats per week = 3/4 = 0.75
The number of weeks that the bag of 3lbs will last = 4 (3/0.75)
Thus, we can conclude that the shopper's one bag of cat food will last her cat for 4 weeks.
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Completion Question:Consider the problem: A shopper buys cat food in bags of 3 lbs. Her cat eats 3/4lb each week. How many weeks does one bag last?
Find the quotient
Thank u!
The result of the quotient expression as required to be determined in the task content is; 2^(8x + 3).
What is the result of the quotient expression?It follows from the laws of indices that 8 = 2³.
On this note, the given expression; 8^(3x + 1) ÷ 2^x can be solved as follows;
2^( 3 (3x + 1) ) ÷ 2^x
= 2^( 9x + 3 ) ÷ 2^x
= 2^(9x + 3 - x)
= 2^(8x + 3)
Ultimately, the result of the quotient is; 2^(8x + 3).
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The sides of a triangle are 21 ,29 , and 11 . Use the Pythagorean Theorem to determine if the triangle is right, acute, or obtuse. The triangle is because the square of the largest side the sum of the squares of the other two sides.
The triangle is obtuse because 11^2 + 21^2 does not equal 29^2.
The Pythagorean Theorem states that the sum of the squares of the two shorter sides of a triangle must equal the square of the longest side in order for the triangle to be a right triangle.
We can use this theorem to determine the type of triangle by calculating the square of the longest side (29^2), and then comparing it to the sum of the squares of the other two sides (11^2 + 21^2). Since 11^2 + 21^2 does not equal 29^2, the triangle is obtuse.
The Pythagorean Theorem states that the sum of the squares of the two shorter sides of a triangle must equal the square of the longest side in order for the triangle to be a right triangle.
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Solve the following radical equation
√4z²-13z+9+3=3z
To solve the radical equation √4z²-13z+9+3=3z, we need to get rid of the radical sign in order to isolate the variable.
First, we can add 9 to both sides to get:
√4z²-13z+12=3z+9
Then we can square both sides of the equation:
4z²-13z+12 = 9z²+27z+81
Now, we can rearrange the equation to get:
9z²-40z+69 = 0
Now we can use the quadratic formula to find the solutions for z:
z = (40 ± √(40²-4969)) / 2*9
z = (40 ± √(1600-2772)) / 18
z = (40 ± √(-1172)) / 18
Since the square root of a negative number is an imaginary number, this equation has no real solutions, which means that there are no values of z that will make the equation true.
So the radical equation √4z²-13z+9+3=3z doesn't have any solution.
Ellie purchased a laptop for $1,400, but she still owes $600 which she borrowed to purchase the laptop. What is her equity in the case of the laptop?
Which of the four do I pick?
If Ellie purchased a laptop for $1,400, but she still owes $600 which she borrowed to purchase the laptop. Her equity in the case of the laptop is; A. $1400+ $600 = $2000.
How to find her equity?Using this formula to determine her equity
Equity = Cost of laptop + Amount owes
Where:
Cost of laptop = $1,400
Amount owes = $600
Let plug in the formula
Equity = $1,400 + $600
Equity = $2,000
Therefore we can conclude that the correct option is A.
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Translate this sentence into an equation 12 more than Craig’s score is 70 use the variable c to represent his score
Reason:
c = Craig's score
c+12 = twelve more than Craig's score
c+12 = 70 because the key term "is" translates to "equals". As an example, saying "x is 5" means "x = 5".
use the ressult of the simulation to explain why you think the smaple orvoideso r does not provide evidence that the standard deviation of the uice dispened exceeds .05 fluid onces
The result of the simulation does not provide evidence that the standard deviation of the juice dispensed exceeds .05 fluid ounces because the simulation does not indicate any instances where the standard deviation of the juice dispensed was greater than .05 fluid ounces. This indicates that the variability of the juice dispensed was within the acceptable sum range of .05 fluid ounces or less.
The simulation did not indicate any instances where the standard deviation of the juice dispensed was greater than .05 fluid ounces, indicating that the variability of the juice dispensed was within the acceptable sum of range
The result of the simulation does not provide evidence that the standard deviation of the juice dispensed exceeds .05 fluid ounces because the simulation does not indicate any instances where the standard deviation of the juice dispensed was greater than .05 fluid ounces. This indicates that the variability of the juice dispensed was within the acceptable range of .05 fluid ounces or less.
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A randomly selected sample of 1,000 college students was asked whether they had ever used the drug Ecstasy. Sixteen percent (16% or 0.16) of the 1,000 students surveyed said they had. Which one of the following statements about the number 0.16 is correct?Select one:A. It is a randomly chosen numberB. It is a population proportionC. It is a sample proportionD. It is a margin of error
It is a sample size proportion. 0.16 is the proportion of the 1,000 college students that reported using Ecstasy, which is a measure of the sample.
The number 0.16 is a sample proportion. This means that it is the proportion of the sample of 1,000 college students that reported using the drug Ecstasy. A sample proportion is a measure of the sample, not of the population. It is not a randomly chosen number and it is not a margin of error. A margin of error is a measure of how accurate a sample statistic is likely to be, usually expressed as a percentage or a number of standard deviations. A population proportion is the proportion of the total population that holds a certain characteristic or has had a certain experience. Thus, 0.16 is a sample proportion and not a population proportion, margin of error, or randomly chosen number
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a boy has color blindness and has trouble distinguishing between blue and green. there are 75 blue pens and 25 green pens mixed together in a box. given that he picks up a blue pen, there is a 80% chance that he thinks it is a blue pen and a 20% chance that he thinks it is a green pen. given that he picks a green pen, there is an 90% chance that he thinks it is a green pen and a 10% chance that he thinks it is a blue pen. assume that the boy randomly selects one of the pens from the box.
a) There is a 60% probability that the boy picks up a blue pen and recognizes it as blue.
b) There is a 62.5% probability that the boy chooses a pen and thinks it is blue.
c) Given that the boy thinks he chose a blue pen, there is a 96% probability that he actually chose a blue pen.
What is probability?
Probability is a way to gauge how likely something is to happen. Many things are difficult to forecast with absolute confidence. Using it, we can only make predictions about the likelihood of an event happening, or how likely it is. Probability can range from 0 to 1, with 0 denoting an impossibility and 1 denoting a certainty.
The following probabilities are given -
A 75% probability that a pen is blue.
A 25% probability that a pen is green.
If a pen is blue, an 80% probability that the boy thinks it is a blue pen.
If a pen is blue, a 20% probability that the boy thinks it is a green pen.
If a pen is green, a 90% probability that the boy thinks it is a green pen.
If a pen is green, a 10% probability that that the boy thinks it is a blue pen.
a) The probability that the boy picks up a blue pen and recognizes it as blue.
There is a 75% probability that he picks up a blue pen.
There is a 80% probability that he recognizes a blue pen as blue.
So, the equation will be -
P = 0.75 × 0.8
P = 0.6
Therefore, there is a 60% probability.
b) The probability that the boy chooses a pen and thinks it is blue.
There is a 75% probability that he picks up a blue pen.
There is a 80% probability that he recognizes a blue pen as blue.
There is a 25% probability that he picks up a green pen
There is a 10% probability that he thinks a green pen is blue.
So, the equation will be -
P = (0.75 × 0.80) + (0.25 × 0.10)
P = 0.6 + 0.025
P = 0.625
Therefore, there is a 62.5% probability.
c) Given that he thinks he chose a blue pen, the probability that he actually chose a blue pen.
There is a 62.5% probability that he thinks that he choose a blue pen.
There is a 60% probability that he chooses a blue pen and think that it is blue.
So, the equation will be -
P = 0.6/0.625
P = 0.96
Therefore, there is a 96% probability.
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A boy has color blindness and has trouble distinguishing blue and green. There are 75 blue pens and 25 green pens mixed together in a box. Given that he picks up a blue pen, there is a 80% chance that he thinks it is a blue pen and a 20% chance that he thinks it is a green pen. Given that he picks a green pen, there is an 90% chance that he thinks it is a green pen and a 10% chance that he thinks it is a blue pen. Assume that the boy randomly selects one of the pens from the box.
a) What is the probability that he picks up a blue pen and recognizes it as blue?
b) What is the probability that he chooses a pen and thinks it is blue?
c) (Given that he thinks he chose a blue pen, what is the probability that he actually chose a blue pen?
