A new auditorium is built with a foundation in the shape of one-fourth of a circle of radius 50 feet. So, it forms a region bounded by the graph of x^2 + y^2 = 50^2 with x ≥ 0 and y ≥ 0. The following equations are models for the floor and ceiling. Floor: z = x+y/ 5 Ceiling: z = 20 + xy /100 Calculate the volume of the room, which is needed to determine the heating and cooling requirements.

Because the function is one-to-one we know that it is invertible, from that we conclude:

If a function is one-to-one, then the function is invertible in its domain, if we define f⁻¹(x) as the inverse, then we can define two relations:

f( f⁻¹(x)) = x

And that if:

f(x)= y

then:

f⁻¹(y) = x

Here we know that:

f(0) = 7

Then the inverse evaluated in 7 must give 0, this is:

f⁻¹(7) = 0

And the second expression is:

( f(0) )⁻¹

This is not an inverse function, this is the inverse of the function evaluated in 0, then we will get:

( f(0) )⁻¹ = ( 7 )⁻¹ = 1/7.

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Step-by-step explanation:

y = x^2 / z      now triple x and z

y = (3x)^2 / 3z = 9x^2 / 3z = 3 * x^2 / z  <==== this is 3 times the original

    y is tripled

Answer:

Step-by-step explanation:

It will take 50 minutes for the turtle and snail to meet.

because they are all moving at feet per minute we can create a formula

total feet = (turtle feet per minute) + (snail feet per minute)

300 = 5m +1m

combine like terms

300 = 6m

divide both sides by 6

50=m

Answer:

See below.

Step-by-step explanation:

Let's denote the distance the turtle walks by x. Then the distance the snail crawls would be 300 − x.

We can now set up an equation to represent the situation. Since distance = rate × time, we have

x/5 = (300 - x)/1

Solving for x, we get

x = 250

So the turtle walks 250 feet before meeting the snail, and the snail crawls the remaining 50 feet.

To find the time it takes for them to meet, we can use either of the two distances and its corresponding rate:

time = distance/rate

For example, using the turtle's distance

time = 250/5 = 50 minutes

Therefore, it takes 50 minutes for the turtle and the snail to meet.

The measure of angle PQS is of 23º, hence the correct option is given by option C.

The angle addition postulate states that if two angles share a common vertex and a common angle, forming a combination, the measure of the larger angle will be given by the sum of the smaller angles.

In the context of this problem, the angle PQR is the combination of angles PQS and SQR, hence the equation is given as follows:

m < PQR = m < PQS + m < SQR.

Replacing the measures into the equation, the value of x is obtained as follows:

5x + 8 + 2x - 3 = 26

7x = 21

x = 3.

Then the measure of angle PQS is obtained as follows:

PQS = 5(3) + 8 = 15 + 8 = 23º.

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y = 1 arctan(1/2(x-0)) - [tex]\frac{\pi }{3}[/tex], which simplifies to y = arctan(1/2(x-[tex]\frac{\pi }{3}[/tex])). So correct option is B.

An equation is a mathematical statement that uses symbols and mathematical operations to show that two quantities are equal. Equations are used to represent a wide range of relationships and can be used to solve problems and make predictions.

The original equation is y = arctan x. To horizontally compress the graph by 1/2, we need to replace x with 2x. The equation becomes y = arctan(2x).

To translate the graph down by [tex]\frac{\pi }{3}[/tex] units, we need to subtract [tex]\frac{\pi }{3}[/tex] from y. The equation becomes y = arctan(2x) - [tex]\frac{\pi }{3}[/tex].

So far, we have y = arctan(2x) - [tex]\frac{\pi }{3}[/tex]. To match the form y = a arctan(b(x-c)) + d, we need to further transform the equation.

We can write 2x as 1/2(4x), so the equation becomes y = arctan(1/2(4x)) - [tex]\frac{\pi }{3}[/tex].

Comparing this with y = a arctan(b(x-c)) + d, we have a = 1, b = 1/2, c = 0, and d = -[tex]\frac{\pi }{3}[/tex].

Substituting these values, we get y = 1 arctan(1/2(x-0)) - [tex]\frac{\pi }{3}[/tex], which simplifies to y = arctan(1/2(x-[tex]\frac{\pi }{3}[/tex])).

Therefore, the answer is B. y= arctan(1/2(x-[tex]\frac{\pi }{3}[/tex])).

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The complete question is:

The equation y = 1 arctan(1/2(x-0)) -π/3 can be written as y = arctan(1/2(x-π/3)).

What is equation ?

An equation is a mathematical statement that proves the equality of two quantities by using symbols and mathematical procedures. Equations can be used to express a wide variety of relationships, solve issues, and generate predictions.

The first formula is y = arctan x. We must swap out x for 2x in order to horizontally compress the graph by half. y = arctan is the new formula.(2x).

We must deduct π/3  from y in order to scale the graph downward by π/3 units. Y = arctan(2x) -π/3 is the new equation.

y = arctan(2x)-π/3  is what we now have. We need to further alter the equation so that it has the form y = an arctan(b(x-c)) + d.

Since 2x can be written as 1/2(4x), the equation changes to y = arctan(1/2(4x)) -π/3.

We have a = 1, b = 1/2, c = 0, and d = π/3- when y = an arctan(b(x-c)) + d is compared to this.

The result of substituting these numbers is y = 1 arctan(1/2(x-0)) -π/3, which may be written as y = arctan(1/2(x-π/3)).

y= arctan(1/2(x-π/3)), hence the solution is B.

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Answer:

The average daily balance on Perry's credit card for the period beginning October 10 was $714, and the balance at the end of the cycle was $506. To calculate the balance due, we need to know the length of the billing cycle. Assuming a 30-day billing cycle, we can use the average daily balance method to calculate the interest charged. The formula for interest charged is: Average Daily Balance x APR x Days in Billing Cycle / 365. Plugging in the numbers, we get: $714 x 0.08 x 30 / 365 = $4.68. Adding this interest to the ending balance of $506, the total balance due is $510.68.

The proof that ΔABC ≅ ΔDCB (congruent) in a parallelogram is because they are alternate interior angles.

If AABC is a parallelogram, then AB and AC are parallel. Therefore, angle BAC is equal to angle BDC since they are alternate interior angles.

Similarly, angle ABC is equal to angle ADC since they are also alternate interior angles.

Now, using the angle-angle-side (AAS) postulate to show that ΔABC and ΔDCB are congruent.

Since angle BAC = angle BDC and angle ABC = angle ADC, angle A = angle D, angle B = angle C, and side AB = side DC.

Therefore, ΔABC ≅ ΔDCB.

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Image transcribed:

8. Prove ΔABC≅ΔDCB (parallelogram)

The value for x in the given triangle is 36.81. The solution has been obtained by using the concept of trigonometry.

What is trigonometry?

Right-angled triangles, including their sides, angles, and connections, are studied in the branch of mathematics known as trigonometry.

We are given a right angled triangle with perpendicular 32 and base x.

We know that tan θ is the ratio of triangle's perpendicular to its base.

Here, θ = 41°

So, we get

⇒ tan 41° = [tex]\frac{32}{x}[/tex]

⇒ 0.8693 = [tex]\frac{32}{x}[/tex]

⇒ 0.8693x = 32

⇒ x = 36.81

Hence, the value for x in the given triangle is 36.81.

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AB has a length of 15, angle A is approximately 36.87 degrees and angle B is approximately 53.13 degrees.

We can use the Pythagorean theorem to find the length of AB:

AB² = AC² + BC²

AB² = 9² + 12²

AB² = 81 + 144

AB² = 225

AB = √225

AB = 15

So, AB has a length of 15.

To find angle A, we can use the inverse tangent function:

tan(A) = opposite/adjacent = AC/BC = 9/12 = 3/4

A = tan⁻¹(3/4) ≈ 36.87°

So, angle A is approximately 36.87 degrees.

To find angle B, we can use the fact that the three angles in any triangle add up to 180 degrees:

B = 90 - A = 90 - 36.87 ≈ 53.13°

So, angle B is approximately 53.13 degrees.

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Answer:

To find an appropriate trigonometric substitution of the form x = f(t) to simplify the integral, we need to look for expressions involving square roots of the form:

sqrt(a^2-x^2)

sqrt(a^2+x^2)

sqrt(x^2-a^2)

If we have sqrt(a^2-x^2), we can use the substitution x = a sin(t) or x = a cos(t) depending on which one of them makes the expression simpler.

If we have sqrt(a^2+x^2), we can use the substitution x = a tan(t) or x = a sec(t).

If we have sqrt(x^2-a^2), we can use the substitution x = a sec(t) or x = a tan(t).

It is important to keep in mind the trigonometric identities and the Pythagorean theorem to simplify the integrals after substituting.

Step-by-step explanation:

hope its help <:

Answer:

least common denominator = 18

Step-by-step explanation:

The answer is C. 0.6 hours. They will play each game for 0.6 hours, or 36 minutes.

What are combinations ?

In mathematics, combinations are a way of selecting objects from a set without regard to their order. In other words, a combination is a selection of objects where the order of the selection does not matter.

To find out how long Gretchen and Nina will play each game, we need to divide the total time they will play by the number of games they will play.

Total time they will play = 3.6 hours

Number of games they will play = 6

So, the time they will play each game = Total time ÷ Number of games

= 3.6 hours ÷ 6 games

= 0.6 hours

Therefore, the answer is C. 0.6 hours. They will play each game for 0.6 hours, or 36 minutes.

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To determine the profit earned from selling 250 shares worth $4.50 each with a 7% fee charged by the stock broker, we need to calculate the total revenue earned and the total cost incurred.

Total revenue earned from selling 250 shares at $4.50 per share:

Revenue = 250 x $4.50 = $1,125

Total cost incurred for buying and selling the shares, including the 7% fee charged by the broker:

Cost = 2 x 250 x $4.50 x 0.07 = $157.50

Here, we are multiplying the number of shares (250) by the price per share ($4.50) to get the total value of the shares. Then we multiply this by 2, since we need to account for both the buying and selling transactions, and then multiply by the broker fee rate of 7% (0.07) to get the total cost incurred.

Therefore, the profit earned from the sale of 250 shares is:

Profit = Revenue - Cost

Profit = $1,125 - $157.50

Profit = $967.50

Hence, the profit earned from selling 250 shares worth $4.50 each with a 7% fee charged by the stock broker is $967.50.

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Answer:

Step-by-step explanation: 5.) Circular shape.

     6.) Feet.

     7.) Circle.

     8.) The circumference is 9 and the radius is 18.

     9.)  By the way we would need to use The Circumference as 2 and the radius to be 9 to equal 18 for the radius circle.

For the given function  f(x) = ⁿ√p(x), The Domain depends on the value of n and The Domain is all real numbers if n is ODD.

What is the definition of a function?

In mathematics, a function is a relation between a set of inputs (also known as the domain) and a set of possible outputs (also known as the range) with the property that each input is associated with exactly one output.

In other words, a function takes an input value, performs a specific operation or set of operations on it, and produces an output value. It can be represented by a rule, formula, or graph that relates each input to its corresponding output.

Now,

The domain of the function f(x) = ⁿ√p(x) depends on the value of n.

If n is odd, then the function is defined for all real numbers, and the domain is all real numbers.

If n is even, then the function is defined only for non-negative real numbers, and the domain is all non-negative real numbers.

Therefore, the statement "The Domain is all real numbers" is not always true for the function f(x) = ⁿ√p(x). The correct statements are:

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After solving the equations e know that Planet II is 35 million miles away from the sun.

The equals sign is a symbol used in mathematical formulas to denote the equality of two expressions.

An equation is a mathematical statement that contains the symbol "equal to" between two expressions with identical values.

As in 3x + 5 = 15, for example.

There are many different types of equations, including linear, quadratic, cubic, and others.

The three primary forms of linear equations are point-slope, standard, and slope-intercept.

So, let x be the separation between planet I and the sun.

Planet II's distance from the sun is x-30.2.

Planet iii's distance from the sun is equal to x+24.8.

x + x-30.2 + x+24.8 = 190.2

Mix related phrases to find x.

3x - 5.4 = 190.2

3x = 195.6

x = 65.2

65.2 million miles separate planet I from the sun.

Planet II is 35 million miles from the sun or 65.2-30.2.

Planet iii is 90 million miles from the sun (65.2 + 24.8 = miles).

Therefore, after solving the equations e know that Planet II is 35 million miles away from the sun.

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Answer: To find the points where the two skaters might collide, we need to find the values of x and y that satisfy both equations:

y = -3x + 8

y = -2x^2 + 7x

We can set the two equations equal to each other and solve for x:

-3x + 8 = -2x^2 + 7x

This simplifies to:

2x^2 - 10x + 8 = 0

Dividing both sides by 2, we get:

x^2 - 5x + 4 = 0

Factoring the left side, we get:

(x - 1)(x - 4) = 0

So the solutions for x are x = 1 and x = 4.

To find the corresponding values of y, we can substitute these values of x into either equation. Let's use y = -3x + 8:

When x = 1, y = -3(1) + 8 = 5

When x = 4, y = -3(4) + 8 = -4

Therefore, the two skaters might collide at the points (1, 5) and (4, -4).

Step-by-step explanation:

the probability of rolling a 6 and a 4 is 1/36 or 2.78%.

The probability of rolling a 6 and a 4 is calculated by taking the number of outcomes that would result in rolling a 6 and a 4 divided by the total number of possible outcomes. To calculate the probability of rolling a 6 and a 4 on a standard six-sided die, we first need to determine the number of outcomes that would result in rolling a 6 and a 4. There is only one way to roll a 6 and a 4 in that order, since each roll is independent of the other and the order matters.There is only one possible outcome that would result in rolling a 6 and a 4 (6,4). There are a total of 36 possible outcomes when rolling two six-sided cubes (6x6). Therefore, the probability of rolling a 6 and a 4 is 1/36 or 2.78%.

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You need to cut  [tex]\frac{7} {10}[/tex] minutes from the song for it to be exactly 3 1/2 minutes.

The unitary technique can be used tο calculate the value οf a single unit frοm the values οf multiple units and the value οf multiple units frοm the values οf single units.

This methοd is frequently emplοyed in math calculatiοns. Yοu can use this methοd tο sοlve issues cοncerning ratiο and prοpοrtiοn, algebra, geοmetry, etc.

The amount of time needed to cut is the difference between these two mixed fraction:

[tex]$ \Rightarrow 4 \frac{1}{5} - 3 \frac{1}{2}[/tex]

[tex]$ \Rightarrow \frac{21}{5} - \frac{7}{2}[/tex]

The LCM of 5 and 2 is 10

[tex]$ \Rightarrow \frac{42- 35} {10}[/tex]

[tex]$ \Rightarrow \frac{7} {10}[/tex]

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Complete question:

You plan to enter a song writing contest. Your song must be exactly 3 1/2 minutes long. You have a song that lasts for 4 1/5 minutes. How many minutes do you need to cut from the song?

Answer:

This isn't a reasonable scale

Step-by-step explanation:

Using the scale 1 in. = 4 in., the width of the scaled down painting will be:

48 in. ÷ 4 = 12 in.

And the height of the scaled down painting will be:

120 in. ÷ 4 = 30 in.

So the scaled down painting will be 12 in. wide and 30 in. tall. Since Sarah's paper is 12 in. wide and 24 in. tall, the scaled down painting will fit within the paper's width but will be taller than the paper.

Therefore, the scale is not reasonable because the scaled down painting will not fit within the dimensions of Sarah's paper.

Answer:

19.2 ft

Step-by-step explanation:

Okay so hopefully. I got the equation right. But I believe this is it

Answer:

Step-by-step explanation:

The equation for the cost of parking in Garage A for t hours is:

A = 3t + 7

The equation for the cost of parking in Garage B for t hours is:

B = 4t + 3

To find the number of hours parked, t, that would make the cost of each garage the same, we can set the two equations equal to each other and solve for t:

3t + 7 = 4t + 3

Subtracting 3t from both sides, we get:

7 = t + 3

Subtracting 3 from both sides, we get:

4 = t

Therefore, if Bella parks for 4 hours, the cost of parking in Garage A and Garage B will be the same.

To verify, we can substitute t = 4 into the two equations:

A = 3(4) + 7 = 19

B = 4(4) + 3 = 19

So, if Bella parks for 4 hours, the cost of parking in Garage A and Garage B will be $19.

Answer:

Bella is deciding between two parking garages. Garage A charges an initial fee of $7 to

park plus $3 per hour. Garage B charges an initial fee of $3 to park plus $4 per hour.

Let A represent the amount Garage A would charge if Bella parks for t hours, and let

B represent the amount Garage B would charge if Bella parks for t hours. Write an

equation for each situation, in terms of t, and determine the hours parked, t, that

would make the cost of each garage the same.

A = $7 + $3

B = $3 + $4

Let x be the number of pounds needed for 6035 square feet.

We can set up a proportion between the pounds of grass seed and the square feet covered:

5 pounds / 355 square feet = x pounds / 6035 square feet

To solve for x, we can cross-multiply and simplify:

5 pounds * 6035 square feet = 355 square feet * x pounds

30175 = 355x

x = 30175 / 355

x ≈ 85.07

Therefore, approximately 85.07 pounds of grass seed are needed for 6035 square feet

To find the range of a data set from a box and whisker plot, you can use the whiskers of the plot. The whiskers represent the minimum and maximum values of the data set, so the range can be calculated by subtracting the minimum value from the maximum value.

A whisker plot, also known as a box and whisker plot, is a graphical representation of the distribution of a set of data. It displays a summary of the minimum, maximum, median, and quartiles of the data set.

A typical whisker plot consists of a rectangular box with a vertical line inside, which represents the median of the data set. The lower and upper edges of the box represent the first and third quartiles, respectively. The length of the box thus represents the interquartile range (IQR), which is a measure of the spread of the middle 50% of the data.

In statistics, the range of a data set can be found from a box and whisker plot by looking at the endpoints of the whiskers.

The whiskers in a box and whisker plot extend from the box to the smallest and largest data points within a certain range of the median. The range of the data set is simply the difference between the smallest and largest data points.

To find the range of the data set from a box and whisker plot, simply identify the endpoints of the whiskers and find the difference between them. Note that the whiskers may be labeled with the actual values of the data points or labeled with a multiple of the interquartile range (IQR), which is the distance between the first and third quartiles of the data set.

It's worth noting that the range is a very basic measure of the spread or dispersion of a data set and is sensitive to outliers. Other measures of spread, such as the interquartile range, variance, or standard deviation, may provide more robust and informative summaries of the data.

Hence, the method to plot whiskers in a data range is provided.

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Answer:

9 seconds

Step-by-step explanation:

right side of the graph touches the x axis at 9

By rewritting the exponential equation, we can see that the correct options are B and C.

We know that the balance is modeled by the exponential equation below:

[tex]A = 328.23\times e^{0.045*(t - 2)}[/tex]

Now we want to see which of the other equations are equivalent to this one, so we need to rewrite this equation, so let's do that.

First we can rewrite the second part to get:

[tex]A = 328.23\times e^{0.045\times(t - 2)}\\\\A = 328.23\times(e^{-2*0.045*}\times e^{0.045\times t})\\\\A = 300\times e^{0.045\times t}[/tex]

So that is an equivalent equation.

We also can keep rewritting this to get:

[tex]A = 300\times e^{0.045\times t}\\\\A = 300\times(e^{0.045})^t\\\\A = 300\times(1.046)^t[/tex]

The correct options are B and C.

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