Please show your work so we can see how you got your answers
1. You have a "small clean room" that is 144" x 144" square with a ceiling height of 120" and 6 HEPA filters in the ceiling that deliver 600 cfm each and 1 access door that is currently closed.
For the above room, answer the following questions;
• What is the area of the room in Sq. Ft?
• What is the volume of the room in Cu. Ft? • What is the Air Change (AC's) Rate if the equation for Air Change Rate is N=60Q/vol
N = # AC’s per hour
Q = Total CFM
Vol = Space volume in cubic ft.
• If the door to this room were now opened, would the room differential pressure with respect to the adjacent hallway, be;
o Positive
o Negative
• Neutral

Answer:

To find the number of batches of granola bars the factory made, we need to divide the total amount of raisins used by the number of raisins used per batch.

First, we need to convert the mixed numbers to improper fractions:

4 5/6 = 29/6

9 2/3 = 29/3

Next, we can divide:

29/3 ÷ 29/6 = 29/3 x 6/29 = 6

Therefore, the factory made 6 batches of granola bars yesterday.

Answer: First, let's calculate the total cost of all the inputs:

Cost of grape concentrate = 2,450 * $0.07 = $171.50

Cost of granulated sugar = 54 * $0.50 = $27

Cost of lemons = 60 * $0.85 = $51

Cost of yeast tablets = 200 * $0.29 = $58

Cost of nutrient tablets = 250 * $0.11 = $27.50

Cost of water = 2,900 * $0.005 = $14.50

Next, let's calculate the amount of each input that is actually used:

Grape concentrate used = 2,450 * (1 - 0.02) = 2,401 oz

Sugar used = 54 * (1 - 0.10) = 48.6 lbs

Lemons used = 60 * (1 - 0.25) = 45

Yeast tablets used = 200

Nutrient tablets used = 250

Water used = 2,900 oz

Now, we can calculate the total cost of the inputs that are actually used:

Total cost of inputs used = $171.50 + $27 + $51 + $58 + $27.50 + $14.50 = $350.50

Finally, we can calculate the standard cost per gallon:

Standard cost per gallon = Total cost of inputs used / Number of gallons produced

Number of gallons produced = 50

Standard cost per gallon = $350.50 / 50 = $7.01 per gallon

Therefore, the standard cost per gallon of wine is $7.01.

Step-by-step explanation:

The equation that shows how to find a percentage is:

part/whole = percent/100

This equation can be used to solve for any of the three variables, given the values of the other two.

None of the given numbers make the equation 8/y² + 2 true

What is algebra?

Algebra is a branch of mathematics that deals with mathematical operations and symbols used to represent numbers and quantities in equations and formulas. It involves the study of variables, expressions, equations, and functions.

To solve this problem, we can substitute each of the given numbers (0, 1, 2, 3, 4) for y in the equation 8/y² + 2 and see if the equation is true.

Substituting y=0 would make the denominator of the fraction zero, which is undefined, so y=0 is not a valid choice.

Substituting y=1 would give us:

8/1² + 2 = 8 + 2 = 10

So, 1 is not the answer.

Substituting y=2 would give us:

8/2² + 2 = 8/4 + 2 = 2 + 2 = 4

So, 2 is not the answer.

Substituting y=3 would give us:

8/3² + 2 = 8/9 + 2 = 0.888 + 2 = 2.888

So, 3 is not the answer.

Substituting y=4 would give us:

8/4² + 2 = 8/16 + 2 = 0.5 + 2 = 2.5

So, 4 is not the answer.

Therefore, none of the given numbers make the equation 8/y² + 2 true.

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Answer:

The scale factor is the ratio of the corresponding side lengths in the original figure and the image.

If the image was created by reducing the size of the original figure, then the scale factor is less than 1.

If the image was created by enlarging the size of the original figure, then the scale factor is greater than 1.

The given options for the scale factor are "one half" and "two fourths".

"One half" is equivalent to 0.5.

"Two fourths" is equivalent to 0.5 as well (since 2/4 can be simplified to 1/2, which is 0.5).

Therefore, the scale factor used to create the image is 0.5 or one half, which suggests that the image was created by reducing the size of the original figure by a factor of 0.5.

Using Pythagorean theorem, the height of Oscar's dog house, at its tallest point, is approximately 7.81 feet.

What is Pythagorean Theorem?

The Pythagorean Theorem is a fundamental concept in mathematics that relates to the sides of a right triangle. It states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. The theorem is expressed mathematically as:

a² + b² = c²

Now,

Let's height of the house = "h":

Using Pythagoras

a² + b² = c²

Where "a" and "b" are the lengths of the slanted sides and "c" is the height of the house. We know that "a" and "b" are both 5 feet long, and the bottom of the house is 6 feet across. Let's use this information to find "c":

a = b = 5 feet

b = 6 feet

c² = a² + b²

c² = 5² + 6²

c² = 25 + 36

c² = 61

c = √(61)

c ≈ 7.81 feet

So the height of Oscar's dog house, at its tallest point, is approximately 7.81 feet.

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Answer:

14.4

Step-by-step explanation:

percentage for corn =30

total no. of acres of the land=48

=30 × 48

100

=14.4

We can use the procedures below to calculate what proportion of the original principal the overall amount of interest represents. This total amount of interest received corresponds to about 5.34% of the initial investment.

Divide the principal even by rate of interest, the time period, and other factors to arrive at simple interest. Simple return Equal principal + interests + hours is the marketing formula.The most typical technique to figure out interest is to use a portion of the principal sum. He would only pay his share of the 100% interest, for example, if somebody borrows $100 from the a partner and pledges to repay the loan with 5% interest. $x (0.05) = $5. When, you must pay interest.when you lend money after borrowing it and adding interest. Interest is often determined as an indicator of the overall of the loan total. The interest rate of the loan is the name given to this percentage.

Determine the total interest that was earned in Step 1.

It states that $33.75 was earned in interest overall.

Step 2: Determine the interest generated before to taxes.

The sum of the interest earned after tax can be determined by dividing the entire sum of the interest by (1 + rate of taxation), where the rate of tax is given as a decimal. Daniela was subject to a tax of 15% on the investment earnings. The tax rate in this instance is 15%, which really is equal to 0.15.

Interest gained before taxes is equal as $33.75 / (1 Plus 0.15), that results in a value of $29.35.

3. Determine the initial principal.

The $550 that Daniela first put into the CD is referred to as the original primary.

Compute the proportion of the initial principle in step four.

By dividing the sum of interest generated after tax by the original principal and multiplying the result by 100 as express it as a percentage, one can determine what proportion of the original principal the entire amount of interest represents.

(Interest paid before taxation / Original principal) / 100 equals the percentage of the original principal.

= ($possess / $550) w x 100

≈ 5.34%

Hence, the total interest earned is equivalent to roughly 5.34% of the initial capital.

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The distance between city A and city B is approximately 442.3 km.

What is the law of cosine?

The Law of Cosines, also known as the Cosine Rule, is a mathematical formula that relates the lengths of the sides of a triangle to the cosine of one of its angles. Specifically, it states that:

c² = a² + b² - 2ab cos(C).

We can use the Law of Cosines to find the distance between City A and City B. Let's call this distance d.

From the information given, we know that:

The distance between the satellite and city A is 450 km.

The distance between the satellite and city B is 340 km.

The angle between city A, the satellite, and city B is 1.5 degrees.

Using the Law of Cosines, we have:

d² = 450² + 340² - 2(450)(340)cos(1.5)

d² = 202500 + 115600 - 2(450)(340)cos(1.5)

d² = 318100 - 122328.8

d² = 195771.2

d = √195771.2

d ≈ 442.3

Therefore, the distance between city A and city B is approximately 442.3 km (rounded to the nearest tenth).

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The width of the rectangle is 4 units.

What is rectangle?

Rectangle is a four sided polygon or specifically a particular type of parallelogram having two opposite sides are equal and one angle is right angle that is 90°. It has four vertices and two diagonals intersect each other.

Given that,

The length of a rectangle is 2 units more than the width.

Let, the width of the rectangle is w units.

Then the length of the rectangle is w+2 units.

Area of any rectangle is length × width

                                        = (w+2)×w square units

Given that,

The area of the rectangle is   24 square units.

Equating both  the values we get,

                    w(w+2)= 24

We have to solve the equation for w.

    Multiplying the bracket term with w we get,

w²+ 2w = 24

⇒ w² + 2w- 24=0

⇒ (w+6)(w-4)=0

so either w= -6 or w=4

As width cannot be negative so w= 4.

Hence, the width of the rectangle is 4 units.

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The mountain has a total height of 6520 feet

Let the height of the mountain be denoted by h, and let x be the distance from the first observation point to the base of the mountain.

From the first observation point, we can form a right triangle with the mountain top and the point of observation, with angle of elevation 30 degrees.

Therefore, we have:

tan(30) = h/(x + 1500)

From the second point, we have

tan(34) = h/x

So, we have

h = x * tan(34)

This gives

tan(30) = x tan(34)/(x + 1500)

So, we have

tan(30) * (x + 1500) = x * tan(34)

Evaluate

0.58 * (x + 1500) = x * 0.67

This gives

x = 9667

Recall that

h = x * tan(34)

So, we have

h = 9667 * tan(34)

Evaluate

h = 6520

Hence, the height is 6520 feet

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Answer:

a) To solve the differential equation y''-2y'-3y= e^4x, we first find the characteristic equation:

r^2 - 2r - 3 = 0

Factoring, we get:

(r - 3)(r + 1) = 0

So the roots are r = 3 and r = -1.

The general solution to the homogeneous equation y'' - 2y' - 3y = 0 is:

y_h = c1e^3x + c2e^(-x)

To find the particular solution, we use the method of undetermined coefficients. Since e^4x is a solution to the homogeneous equation, we try a particular solution of the form:

y_p = Ae^4x

Taking the first and second derivatives of y_p, we get:

y_p' = 4Ae^4x

y_p'' = 16Ae^4x

Substituting these into the original differential equation, we get:

16Ae^4x - 8Ae^4x - 3Ae^4x = e^4x

Simplifying, we get:

5Ae^4x = e^4x

So:

A = 1/5

Therefore, the particular solution is:

y_p = (1/5)*e^4x

The general solution to the non-homogeneous equation is:

y = y_h + y_p

y = c1e^3x + c2e^(-x) + (1/5)*e^4x

b) To solve the differential equation y'' + y' - 2y = 3xe^x, we first find the characteristic equation:

r^2 + r - 2 = 0

Factoring, we get:

(r + 2)(r - 1) = 0

So the roots are r = -2 and r = 1.

The general solution to the homogeneous equation y'' + y' - 2y = 0 is:

y_h = c1e^(-2x) + c2e^x

To find the particular solution, we use the method of undetermined coefficients. Since 3xe^x is a solution to the homogeneous equation, we try a particular solution of the form:

y_p = (Ax + B)e^x

Taking the first and second derivatives of y_p, we get:

y_p' = Ae^x + (Ax + B)e^x

y_p'' = 2Ae^x + (Ax + B)e^x

Substituting these into the original differential equation, we get:

2Ae^x + (Ax + B)e^x + Ae^x + (Ax + B)e^x - 2(Ax + B)e^x = 3xe^x

Simplifying, we get:

3Ae^x = 3xe^x

So:

A = 1

Therefore, the particular solution is:

y_p = (x + B)e^x

Taking the derivative of y_p, we get:

y_p' = (x + 2 + B)e^x

Substituting back into the original differential equation, we get:

(x + 2 + B)e^x + (x + B)e^x - 2(x + B)e^x = 3xe^x

Simplifying, we get:

-xe^x - Be^x = 0

So:

B = -x

Therefore, the particular solution is:

y_p = xe^x

The general solution to the non-homogeneous equation is:

y = y_h + y_p

y = c1e^(-2x) + c2e^x + xe^x

c) To solve the differential equation y" - 9y' + 20y = x^2*e^4x, we first find the characteristic equation:

r^2 - 9r + 20 = 0

Factoring, we get:

(r - 5)(r - 4) = 0

So the roots are r = 5 and r = 4.

The general solution to the homogeneous equation y" - 9y' + 20y = 0 is:

y_h = c1e^4x + c2e^5x

To find the particular solution, we use the method of undetermined coefficients. Since x^2*e^4x is a solution to the homogeneous equation, we try a particular solution of the form:

y_p = (Ax^2 + Bx + C)e^4x

Taking the first and second derivatives of y_p, we get:

y_p' = (2Ax + B)e^4x + 4Axe^4x

y_p'' = 2Ae^4x +

Answer:

slope of every parallel line is - 4

Step-by-step explanation:

calculate the slope m of the given line using the slope formula

m = [tex]\frac{y_{2}-y_{1} }{x_{2}-x_{1} }[/tex]

with (x₁, y₁ ) = (- 2, 7) and (x₂, y₂ ) = (1, - 5) ← 2 points on the line

m = [tex]\frac{-5-7}{1-(-2)}[/tex] = [tex]\frac{-12}{1+2}[/tex] = [tex]\frac{-12}{3}[/tex] = - 4

• Parallel lines have equal slopes

Thus the slope of any line parallel to the given line is m = - 4

Answer: The collection of whole numbers greater than one trillion below is well defined and is a set. It consists of all whole numbers greater than 1 trillion and less than infinity. Although the set may be infinite, it is still well defined and can be defined using set-builder notation as:

{ x ∈ ℤ | 1,000,000,000,000 < x < ∞ }

or using interval notation as:

(1,000,000,000,000, ∞)

Step-by-step explanation:

This means that he can take up to 5.5 minutes to decorate each cookie and still have enough time to prepare the gift bags within the 90-minute time limit.

In mathematics, an inequality is a statement that compares two values or expressions, indicating whether they are equal, not equal, greater than, less than, greater than or equal to, or less than or equal to each other. Inequalities are represented using symbols such as < (less than), > (greater than), ≤ (less than or equal to), and ≥ (greater than or equal to).

Here,

To determine how many minutes Winston can take to decorate each cookie, we need to use an inequality that takes into account the total time available and the time it takes to decorate each cookie and prepare each gift bag.

Let x be the number of minutes it takes Winston to decorate each cookie.

The total time Winston has available is 90 minutes, and he needs to decorate 12 cookies and prepare 12 gift bags. The time it takes him to prepare each gift bag is 2 minutes. Therefore, the total time it takes him can be expressed as:

Total time = Time to decorate 12 cookies + Time to prepare 12 gift bags

Total time = 12x + 12(2)

Total time = 12x + 24

To determine the maximum time Winston can take to decorate each cookie, we need to find the largest value of x that satisfies the total time constraint. Since he has exactly 90 minutes, we can use the inequality:

12x + 24 ≤ 90

Subtracting 24 from both sides, we get:

12x ≤ 66

Dividing both sides by 12, we get:

x ≤ 5.5

Therefore, the inequality that can be used to determine how many minutes Winston can take to decorate each cookie is:

x ≤ 5.5

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Answer:

Step-by-step explanation:

a lot (90,000) (i think)

The length of line segment AB is equal to 8 units.

The magnitude of m∠BAQ is equal to 58°.

In Mathematics and Geometry, a perpendicular bisector can be defined as a line that bisects or divides a line segment exactly into two (2) equal halves and forms an angle that has a magnitude of 90 degrees at the point of intersection.

This ultimately implies that, the length of line segment AB can be calculated by using the following mathematical equation;

RP = 2AB

AB = RP/2

AB = 16/2

AB = 8 units.

Since A and B are the midpoints of PQ and RQ respectively, we have the following angles;

m∠P + m∠Q + m∠R = 180° (sum of all interior angles of ∆PQR)

58° + 38° + m∠R = 180°

m∠R = 180° - (58° + 38°)

m∠R = 84°

Since PR || AB, we have;

m∠R = m∠ABQ = 84° (corresponding angles).

m∠Q + m∠ABQ + m∠BAQ = 180° (sum of all interior angles of ∆ABQ).

m∠BAQ = 180° - (84° + 38°)

m∠BAQ = 180° - 122°

m∠BAQ = 58°.

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Complete Question:

In the ΔPQR, A and B are the midpoints of PQ and RQ respectively. If RP = 16, m∠P = 58°, and m∠Q = 38°, obtain AB and m∠BAQ.

Answer:

B

Step-by-step explanation:

the average rate of change of h(x) in the interval [ a, b ] is

[tex]\frac{h(b)-h(a)}{b-a}[/tex]

here the [ a, b ] = [ 4, 8 ] , then

h(b) = h(8) = 3 ← point (8, 3 ) on graph

h(a) = h(4) = 9 ← point (4, 9 ) on graph

then

average rate of change = [tex]\frac{3-9}{8-4}[/tex] = [tex]\frac{-6}{4}[/tex] = - [tex]\frac{3}{2}[/tex]

Answer: 50 Dollars

Step-by-step explanation:

Minimum pay: $50 / week

Bonus: $7.5 or more / Potential Customer

The least amount of money you can earn after contacting eight businesses in 1 week is $50, since there is a possibility none of the eight businesses you contacted is a potential customer.

Feel free to correct me if I'm wrong :)

Answer: d=10

Step-by-step explanation:

Answer:

Let x be the amount of the short-term note and y be the amount of the long-term note.

We know that the total amount of the two notes is $160,000:

x + y = 160,000

We also know that the total annual interest paid is $15,350:

0.11x + 0.08y = 15,350

To solve for x and y, we can use the first equation to solve for one variable in terms of the other:

y = 160,000 - x

Substitute this expression for y in the second equation:

0.11x + 0.08(160,000 - x) = 15,350

Simplify and solve for x:

0.11x + 12,800 - 0.08x = 15,350

0.03x = 2,550

x = 85,000

Substitute this value for x in the equation y = 160,000 - x to find y:

y = 160,000 - 85,000 = 75,000

Therefore, the amount of the short-term note is $85,000 and the amount of the long-term note is $75,000.

The value of Y is 54 degrees

A pentagon is described as any five-sided polygon or 5-gon that has a sum of the internal angles in a simple pentagon is 540°.

we know that every side of a regular pentagon is same.

So we have an isosceles triangle. The  correspond angle of y, must equal to y (degree) too.

we also have that every inner angle of a regular pentagon is 360/5=72.

So we can calculate angle y, by the equation y+y+72=180.

Therefore y=54

In conclusion,  the regular pentagon each interior angle measures 108°, and each exterior angle measures 72°

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Complete question is attached in image;

Answer:

x = 58.03°

Step-by-step explanation:

Given:

Solve for x:

1. [tex]x=cos^{-1}(\frac{9}{17})=58.03[/tex]

Answer:

So, the measure of angle x is equal to 58.03.

Using factorization and simplifying the equations, the points of intersections are (-2, 0), ( [ -1 - 3√(7) ] / 2, 4[ -1 - 3√(7) ] / 2 - 11 ) and ( [ -1 + 3√(7) ] / 2, 4[ -1 + 3√(7) ] / 2 - 11 )

We are given two equations:

y = 4x² - 3x + 3

y = x³ + 7x² - 3x + d

and we know that they intersect at x = -4, so we can substitute -4 for x in both equations:

y = 4(-4)² - 3(-4) + 3 = 49

y = (-4)³ + 7(-4)² - 3(-4) + d = -64 + 112 + 12 + d = 60 + d

So, at x = -4, we have y = 49 and y = 60 + d. Since the graphs intersect, these two equations must be equal:

49 = 60 + d

Solving for d, we get:

d = -11

Therefore, the two equations become:

y = 4x² - 3x + 3

y = x³ + 7x² - 3x - 11

We can now set them equal to each other:

4x² - 3x + 3 = x³ + 7x² - 3x - 11

Simplifying and rearranging, we get:

x³ + 3x² - 8x - 14 = 0

We can try to factor this expression by testing possible roots. One possible root is x = 2, because if we substitute 2 for x, we get:

2³ + 3(2)² - 8(2) - 14 = 8 + 12 - 16 - 14 = -10

Since this expression evaluates to a non-zero value, x = 2 is not a root. Similarly, we can test x = -1:

(-1)³ + 3(-1)² - 8(-1) - 14 = -1 + 3 + 8 - 14 = -4

This expression also evaluates to a non-zero value, so x = -1 is not a root. Finally, we can test x = -2:

(-2)³ + 3(-2)² - 8(-2) - 14 = -8 + 12 + 16 - 14 = 6

This expression evaluates to zero, so x = -2 is a root. Using long division or synthetic division, we can divide the cubic polynomial by x + 2 to get:

x³ + 3x² - 8x - 14 = (x + 2)(x² + x - 7)

The quadratic factor x² + x - 7 can be factored using the quadratic formula, giving us:

x² + x - 7 = [ -1 ± √(1 + 4*7) ] / 2

= [ -1 ± 3√(7) ] / 2

Therefore, the three intersection points are:

(-2, 0)

( [ -1 - 3√(7) ] / 2, 4[ -1 - 3√(7) ] / 2 - 11 )

( [ -1 + 3√(7) ] / 2, 4[ -1 + 3√(7) ] / 2 - 11 )

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Answer:

35 pears

Step-by-step explanation:

To make the ratio of apples to pears in the fruit basket 2:3, we need to find out how many pears we need to add.

The current ratio of apples to pears is 12:7. To make it 2:3, we need to multiply both sides of the ratio by a common factor that will transform 12 into 2 and 7 into 3.

Let's find that common factor:

12 : 2 = 6

7 : 3 = 2.333...

Since 12 can be transformed into 2 by dividing by 6, and 7 can be transformed into 3 by multiplying by 2, the common factor is 6.

Now, let's multiply both sides of the original ratio by 6:

12 * 6 : 7 * 6 = 72 : 42

So, the ratio of apples to pears after adding the appropriate number of pears to the basket to make it 2:3 would be 72:42.

To find out how many pears we need to add, we can subtract the current number of pears (7) from the desired number of pears (42):

42 - 7 = 35

Therefore, we need to add 35 pears to the basket to make the ratio of apples to pears 2:3.

The solution to the system of linear equations is (x,y) = (0,1).

A linear equation is an algebraic equation in which the highest power of the variable is one. It represents a straight line when graphed on a coordinate plane.

Linear equations are used to model many real-world situations, such as distance versus time, cost versus quantity, and temperature versus time. They are also used to solve problems in mathematics, physics, engineering, economics, and many other fields.

The slope of a line represents the rate of change of y with respect to x, and can be calculated by dividing the change in y by the change in x between any two points on the line.

In summary, a linear equation is a mathematical expression that represents a straight line and is used to model and solve problems in many fields.

To solve this system of linear equations:

1/3x + y = 1 --------(1)

2x + 6y = 6 --------(2)

We can use the elimination method, which involves adding or subtracting the equations to eliminate one of the variables.

Multiplying equation (1) by 6, we get:

2x + 6y = 6

Now we can subtract equation (1) from this to eliminate y:

(2x + 6y) - (2x + 2y) = 6 - 2

4y = 4

y = 1

Substituting y = 1 into equation (1), we get:

1/3x + 1 = 1

1/3x = 0

x = 0

Therefore, the solution to the system of linear equations is (x,y) = (0,1).

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Answer:

data given

half life of Ra is 1590 years

time for decay is 2000 years

initial amount 300mg

Required final amount

Step-by-step explanation:

from

nt/no=(1/2)^(t/t1/2)

where

nt is final amount

no is initial amount

t is time for decay

t1/2 is half life

now,

nt/300=(1/2)^(2000/1590)

nt/300=(1/2)^1.26

nt/300=0.42

nt= 0.42×300

nt=126mg

: .it will remain 126gm

The equation of the Giant Wheel is: x²+ y² - 150y + 1656 = 0

A circle is a closed, two-dimensional object where every point in the plane is equally spaced from a central point. The line of reflection symmetry is formed by all lines that traverse the circle. Additionally, every angle possesses rotational symmetry around the centre.

The group of points whose separation from a fixed point has a constant value are represented by a circle. The radius of the circle, abbreviated r, is a constant that describes this fixed point, which is known as the circle's centre. The usual equation for a circle with a centre at (x 1, y 1 ) and a radius of r is ( x- x 1 )² + ( y- y 1 )² = r²

We can start by finding the center of the circle. Since the circle is centered on the y-axis, we know that the x-coordinate of the center is 0. To find the y-coordinate, we can use the fact that the platform is 12 feet tall and the overall height of the wheel is 138 feet. This means that the center of the circle is 12 + 63 = 75 feet above the ground, so the y-coordinate of the center is 75.

Next, we can use the formula for the equation of a circle centered at (h, k) with radius r:

(x - h)² + (y - k)² = r²

Since the center of the circle is at (0, 75) and the diameter is 126 feet, the radius is 63 feet. Substituting these values into the formula, we get:

x² + (y - 75)² = 63²

Simplifying, we have:

x² + y² - 150y + 5625 = 3969

x² + y² - 150y + 1656 = 0

So the equation of the Giant Wheel is:

x² + y² - 150y + 1656 = 0

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a. The decision rule for this hypothesis test is: Reject H0 if the test statistic z is less than -1.645 or greater than 1.645. b. The test statistic for this hypothesis test is:  -2.05.

In statistics, a hypothesis is an assumption or claim about a population parameter that can be tested by collecting and analyzing sample data.

According to given information:

a. The decision rule for this hypothesis test is:

Reject H0 if the test statistic z is less than -1.645 or greater than 1.645.

b. The sample proportion of customers who prefer black teas is:

= 0.41

The population proportion under the null hypothesis is:

p = 0.51

The sample size is:

n = 158

The test statistic for this hypothesis test is:

[tex]z = ( - p) / \sqrt(p * (1 - p) / n)[/tex]

[tex]= (0.41 - 0.51) / \sqrt(0.51 * 0.49 / 158)[/tex]

[tex]= -2.05[/tex]

c. The test statistic z of -2.05 is less than the critical value of -1.645, so we can reject the null hypothesis H0. At the 0.10 significance level, there is enough evidence to suggest that the proportion of customers who prefer black teas is not 0.51, and the manager of Tea for Us should consider adjusting their stock orders accordingly.

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