The scale factor of the blueprint of a gymnasium to the actual gymnasium is 1in/15ft. The area of the flow on the blueprint is 114in2.
What is the area of the actual gymnasium?

Answer:

a) [-1, ∞-) and [1, ∞+)

b)  (√2/2,∞)  and (0,-√2/2)

c) y=0, y=2 and y=4

Step-by-step explanation:

a) On what intervals is f increasing?  

We need to proceed some steps, to answer it properly. What are the critical points? So  

1) Differentiate the original function

[tex]f(x)=3x^{5}-5x^3+2\\ f'(x)=15x^4-15x^{2}[/tex]

2) Turn this into an equation  

[tex]15x^4-15x^2=0\\ u=x^{2} and \\u^{2}=x^{4} \\ Then\\ 15u^{2} -15u=0\\ \\15u(u^2 -1)=0\\Therefore\\u=0,u=1[/tex]

Substitute back to x

[tex]x^{2} =u\\ x^{2} =u^4\\ x^{2}=1\\ x= 1,x=-1\\ x^{2} =0\\ x=0[/tex]

Those are x-coordinates of the critical points of [tex]f(x)=3x^{5}-5x^3+2[/tex]

Using the critical points x, to find y-coordinates of those critical points: namely, maximum point, saddle point and minimum point

x=-1 [tex]y=3x^{5}-5x^3+2[/tex]

[tex]y=3(-1)^{5}-5(-1)^3+2[/tex]

y=4

(-1,4) Maximum point

--

x=0 [tex]y=3x^{5}-5x^3+2[/tex]

[tex]y=3(0)^{5}-5(0)^3+2[/tex] y=2

(0,2) Saddle Point

--

x=1 [tex]y=3x^{5}-5x^3+2[/tex]

[tex]y=3(1)^{5}-5(1)^3+2[/tex]  

(1,0) Minimum point

f(x) increases when any value of  x ∈ (-∞, -1] and x ∈ [1, ∞+)  is plugged in the function.  Or we can rewrite as  x[tex]x\leq -1\\ \\x\geq 1[/tex]

-------------------------------------

b)On what intervals is the graph of f concave upward?

To find out which intervals are these, we need to calculate the 2nd derivative.

[tex]f(x)=3x^{5}-5x^3+2\\ f(x)'=15x^4-15x^{2} \\f(x)''=60x^3-30x\\[/tex]

Since we want the concave upward

f(x)''>0

Then:

[tex]60x^3-30x>0\\ 30x(2x^2-1)>0\\[/tex]

Rewriting

[tex]30x(\sqrt{2x}-1)[/tex]

Then

x=0, x=[tex]\frac{\sqrt{2}}{2} \\-\frac{\sqrt{2}}{2}[/tex]

Studying the sign

The intervals >0, when the graph is concave upwards.

are (√2/2,∞)  and (0,-√2/2)

c) Write the equation of each horizontal tangent line to the graph of f.

Generally, every time somebody ask for a tangent line of any given function, they provide an information. A specific point.

But when they do not? Like in this case?

We have to look for where the 1st function's derivative is zero.

As we 've done previously in letter a. x=-1, x=0, x=1

Let's plug them back to the original function

[tex]y=3x^{5}-5x^{3}+2\\   y=3(-1)^{5}-5(-1)^{3}+2\\ y=-3-5+2\\ y=-8+2\\ y=4\\ \\y=3x^{5}-5x^{3}+2\\ y=3(0)^{5}-5(0)^{3}+2\\ y=2\\\\ y=3x^{5}-5x^{3}+2\\ y=3(1)^{5}-5(1)^{3}+2\\ y=0[/tex]

So each horizontal tangent line, tangents the maximum, saddle and the minimum point.

An X-intercept in a rational function is the point where the graph intersects the x-axis. It is derived by setting the function equal to zero and then solving for the variable x. This solution, x, represents the x-intercept of the rational function.

In Mathematics, the x-intercept of a rational function is the point where the function intersects the x-axis. The x-intercept(s) can be found by setting the function equal to zero and solving for x. When graphing, this point determines where the line crosses the x-axis.

The same principle is applied to the calculation of the x-intercept as other functions. The given rational function is set equal to zero, and then algebraic methods are used to solve for the value of x. For example, if given the rational function y = 2x/3x + 1, setting this function to zero would read as 0 = 2x/(3x + 1). Cross-multiplication can then be applied to find the solution for x, which represents the x-intercept.

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To find the 60th percentile, multiply 60/100 by the total number of observations plus one, which equates to 4.8. This falls between the 4th and 5th value, and rounding up means the 60th percentile corresponds to the 5th value, which is 47.

To find the value that corresponds to the 60th percentile in the given data set {12, 28, 35, 42, 47, 49, 50}, we need to follow a series of steps.

Therefore, the value corresponding to the 60th percentile in this data set is 47.

Final answer:

The 60th percentile value in the data set {12, 28, 35, 42, 47, 49, 50} is 47, which is the 5th number in the ordered set after calculating the index.

Explanation:

The 60th percentile value in a data set is found by first arranging the data in ascending order, which is already done in the provided data set. Then calculate the index using the formula Index = P(N + 1), where P is the percentile in decimal form and N is the number of data points. In this case, the 60th percentile index for our 7-point data set is 0.60 × (7 + 1) = 4.8. Since the index is not a whole number, we round up to the next whole number, which here would be 5. Therefore, we take the 5th number in the ordered set, which is 47.

Answer:

50 meters

A is correct

Step-by-step explanation:

Given: 0.05 kilometers

As we know, In 1 km = 1000 m

So, multiply 0.05 km to 1000 to change into meter

[tex]\Rightarrow 0.05\ km\times \dfrac{1000\ m}{1\ km}[/tex]

[tex]\Rightarrow \dfrac{5}{100}\times 1000[/tex]

[tex]\Rightarrow 50\ km[/tex]

Hence, In 0.05 km is 50 meters.

The meter equivalent of 0.05 km is 50 .

Given,

0.05 kilometers

As we know, In 1 km = 1000 m

So, multiply 0.05 km to 1000 to change into meter

0.05 * 1000

= 50 m

Hence, In 0.05 km is 50 meters.

Thus option A is correct .

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The product is 3x² - 7x - 6 after combining like terms.

The student asked to multiply the binomials (3x+2) and (x-3). To do this, we use the FOIL method, which stands for First, Outside, Inside, Last. This refers to the elements in each binomial we need to multiply together.

First, we multiply the first terms in each binomial: (3x) and (x), which gives 3x².

Next, we multiply the outside terms: (3x) and (-3), which gives -9x.

Then we multiply the inside terms: (2) and (x), which produces +2x.

Last, we multiply the last terms of each binomial: (2) and (-3), yielding -6.

Finally, we combine the like terms to get the expanded product - 3x² - 7x - 6.

Answer: Which fraction below represents the following statement?“In the third game of the season, Martha made one out of five free throws.” The answer for this questions would be 1/5

Step-by-step explanation: I got it correct on Edge.

The fraction that represents the statement "in the third game of the season Martha made one of 5 free throws is 1/5.

A fraction is written with a numerator and a denominator where the numerator is less than the denominator.

Example: 1/3 is a fraction.

We have,

Total number of free throws = 5

Number of free throws Martha made = 1

The number of free throws Martha made can be written in fraction form as:

= Number of free throws Martha made / Total number of free throws

= 1 / 5

Thus,

The fraction that represents the statement "in the third game of the season Martha made one of 5 free throws is 1/5.

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The value of x in the equation 6 - x = 2(x - 6) is 6. To convert this to feet, the calculation shows that 6 miles is equivalent to 31,680 feet.

To solve the equation 6 - x = 2(x - 6), follow these steps:

So, the value of x is 6.

To convert 6 miles to feet, using the equation 1 mile/5280 feet = 6 miles/x feet, we cross multiply to get:

Therefore, 6 miles is equivalent to 31,680 feet.

Yolanda paid for her movie ticket using 28 coins, all nickels and quarters. The ticket cost $4.

Let n be the number of nickels

and q be the number of quarters

1 nickels = $0.05 , so n nickels cost = 0.05n

1 quarter = $0.25, so q quarters cost = 0.25q

Total number of coins = 28

number of nickels + number of quarters = 28

n + q = 28 ------- equation 1

Total ticket cost =$4

cost of n nickels + cost of q quarters = $4

0.05n + 0.25q = 4 -------> equation 2

system of linear equations can be used to find the number of nickels, n, and the number of quarters, q, Yolanda used are

n + q = 28

0.05n + 0.25q = 4

The standard deviation for the sample is approximately 0.348 hours.

The critical value for a 90% confidence interval is approximately 1.645, and for a 95% confidence interval, it is approximately 1.96.

To calculate the standard deviation for the sample, we use the formula:

[tex]\[ \text{Standard deviation for sample} = \frac{\text{Population standard deviation}}{\sqrt{\text{Sample size}}} \][/tex]

Given the population standard deviation of 2.2 hours and a sample size of 40, we plug these values into the formula:

[tex]\[ \text{Standard deviation for sample} = \frac{2.2}{\sqrt{40}} \][/tex]

[tex]\[ \text{Standard deviation for sample} \approx \frac{2.2}{6.324} \approx 0.348 \text{ hours} \][/tex]

So, the standard deviation for this sample is approximately 0.348 hours.

For the critical values of confidence intervals, we refer to the z-table or use a calculator.

For a 90% confidence interval, the critical value is approximately 1.645, and for a 95% confidence interval, it is approximately 1.96.

These values represent the number of standard deviations from the mean that encompass the specified confidence level.

Final answer:

The rate of change for the chef's cooking scenario is calculated by dividing the lbs of chicken by the number of people for each case. In both situations, the rate of change is 1/4 lb per person.

Explanation:

The question asks to find the rate of change for a situation where a chef cooks a certain amount of chicken for a specific number of people. To calculate the rate of change, we need to divide the amount of chicken by the number of people for each situation, and then compare.

For 9 lbs of chicken for 36 people, the rate is 9 lbs ÷ 36 people = 0.25 lbs/person, which is equal to 1/4 lb per person. For 17 lbs of chicken for 68 people, the rate is 17 lbs ÷ 68 people = 0.25 lbs/person, which is also 1/4 lb per person.

Since the rate of change is the same in both cases, 1/4 lb per person, the correct answer is:

C. 1/4 lb per person

Answer:45

Step-by-step explanation:

Julia can type 30 words per minute.

Julia can type 150 words in five minutes.

To find out how many words she can type in one minute, divide 150 by 5, which equals 30.

[tex]\frac{150}{5} = 30[/tex]

So she can type 30 words per minute.

Answer:

80000

Step-by-step explanation:

The maximum area that can be enclosed with 800 feet of fencing and the river bordering one side is 0 square feet.

To find the maximum area that can be enclosed with 800 feet of fencing, we need to maximize the area of the rectangular plot. A rectangle with one side bordering the river will have two sides of equal length and two sides of different length. Let's denote the length of the side parallel to the river as x and the length of the other side as y. Since there are two sides of equal length, the total length of those two sides would be 2x. The other two sides would be y each, giving a total length of 2y. The equation we can form based on the given information is:

2x + y + y = 800

Simplifying the equation, we get:

2x + 2y = 800

Dividing both sides by 2, we have:

x + y = 400

Now, we need to express one of the variables in the equation in terms of the other variable. Let's solve for x:

x = 400 - y

To find the maximum area, we need to express it in terms of a single variable. The area of a rectangle is given by length multiplied by width. So, the area A can be expressed as:

A = x * y = (400 - y) * y

To find the maximum value of A, we can use calculus. Let's take the derivative of A with respect to y:

A' = -y + 400

Setting A' to 0 and solving for y, we get:

0 = -y + 400

y = 400

Substituting this value back into the expression for A, we have:

A = (400 - 400) * 400 = 0

Therefore, the maximum area that can be enclosed is 0 square feet.

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To determine which system has no solutions, we analyze a set of inequalities to identify the contradictory statement.

To determine which system has no solutions, we need to analyze the given inequalities:

x < 5

x > 1

x < 5

x < 1

x > 5

x > 1

x > 5

x < 1

From the list, the system 'x < 1' and 'x > 1' has no solutions as it creates a contradictory statement.