University officials say that at least 70% of the voting student population supports a fee increase. If the 95% confidence interval estimating the proportion of students supporting the fee increase is [0.75, 0.85], what conclusion can be drawn?
A. 70% is not in the interval, so another sample is needed.
B.70% is not in the interval, so assume it will not be supported.
C.The interval estimate is above 70%, so infer that it will be supported.
D.Since this was not based on population, we cannot make a conclusion.

Answer:

The 99% confidence interval is be given by (4.872;8.628)

Step-by-step explanation:

A confidence interval is "a range of values that’s likely to include a population value with a certain degree of confidence. It is often expressed a % whereby a population means lies between an upper and lower interval".

The margin of error is the range of values below and above the sample statistic in a confidence interval.

Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".

The dataset is:

6​, 9​, 3​, 9​, 6​, 6​, 7​, 7​, 8​, 9​, 3​, 8

2) Compute the sample mean and sample standard deviation.  

In order to calculate the mean and the sample deviation we need to have on mind the following formulas:  

[tex]\bar X= \sum_{i=1}^n \frac{x_i}{n}[/tex]  

The value obtained is [tex]\bar X=6.75[/tex]

[tex]s=\sqrt{\frac{\sum_{i=1}^n (x_i-\bar X)}{n-1}}[/tex]  

The sample deviation obtained is [tex]s=2.094[/tex]

3) Find the critical value t* Use the formula for a CI to find upper and lower endpoints

In order to find the critical value we need to take in count that our sample size n =12 <30 and on this case we don't know about the population standard deviation, so on this case we need to use the t distribution. Since our interval is at 99% of confidence, our significance level would be given by [tex]\alpha=1-0.99=0.01[/tex] and [tex]\alpha/2 =0.005[/tex]. The degrees of freedom are given by:

[tex]df=n-1=12-1=11[/tex]

We can find the critical values in excel using the following formulas:

"=T.INV(0.005,11)" for [tex]t_{\alpha/2}=-3.106[/tex]

"=T.INV(1-0.005,11)" for [tex]t_{1-\alpha/2}=3.106[/tex]

The confidence interval for the mean is given by the following formula:

[tex]\bar X \pm t_{\alpha/2}\frac{s}{\sqrt{n}}[/tex]  

The next step would be calculate the limits for the interval

Lower interval :

[tex]\bar X - t_{\alpha/2}\frac{s}{\sqrt{n}}[/tex]  

[tex]6.75 - 3.106 \frac{2.094}{\sqrt{12}}=4.872[/tex]  

Upper interval :  

[tex]6.75 + 3.106 \frac{2.094}{\sqrt{12}}=8.628[/tex]  

So the 99% confidence interval would be given by (4.872;8.628)

99% of the time, when we calculate a confidence interval with a sample of n=12, the true mean of rate of attractiveness of their female​ dates will be between the 4.872 and 8.628.

Answer:

E. 12 days

Step-by-step explanation:

So first, we need to find the rates at which each machine will produce w widgets. For machine y, its rate would be:

[tex]y=\frac{W}{T}[/tex]

where W is the number of widgets produced and T is the time it takes to produce them.

We know that x takes 2 more days to produce the same amount of widgets, so the time it takes machine x to produce them can be written as T+2. This will give us the following rate for machine x:

[tex]x=\frac{W}{T+2}[/tex]

the problem also tells us that the two machines working together will produce 5W/4 widgets in 3 days, so if we add the rates for x and y, we will get the total rate which would be:

[tex]x+y=\frac{5W/4}{3}[/tex]

which can be simplified to:

[tex]x+y=\frac{5W}{12}[/tex]

we can now substitute the rates for x and y in the equation so we get:

[tex]\frac{W}{T+2}+\frac{W}{T}=\frac{5W}{12}[/tex]

we can simplify this equation by dividing everything into W, so we get:

[tex]\frac{1}{T+2}+\frac{1}{T}=\frac{5}{12}[/tex]

and we can multiply everything by the LCD. In this case the LCD is 12T(T+2) so we get:

[tex]\frac{1}{T+2}(12T)(T+2)+\frac{1}{T}(12T)(T+2)=\frac{5}{12}(12T)(T+2)[/tex]

which simplifies to:

12T+12(T+2)=5T(T+2)

we can do the respective multiplications so we get:

[tex]12T+12T+24=5T^{2}+10T[/tex]

which simplifies to:

[tex]24T+24=5T^{2}+10T[/tex]

and now we can set the equation equal to zero so we end up with:

[tex]5T^{2}+10T-24T-24=0[/tex]

which simplifies to:

[tex]5t^{2}+14T-24=0[/tex]

now we can solve this by any of the available methods there are to solve quadratic equations. I will solve it by factoring, so we get:

(5T+6)(T-4)=0

so we can set each of the factors equal to zero so we get:

5T+6=0

[tex]T=-\frac{6}{5}[/tex]

this answer isn't valid because there is no such thing as a negative time. So we find the next time then:

T-4=0

T=4

So it takes 4 days for machine x to produce W widgets. We can now rewrite x's rate like this:

[tex]x=\frac{W}{T+2}[/tex]

so

[tex]x=\frac{W}{4+2}[/tex]

[tex]x=\frac{W}{6}[/tex]

With this information, we know that the number of wigets produced can be found by using the following formula:

W=xd

in this case d is the number of days (this is for us not to confuse the previous T with the new time)

so when solving for d we get that:

[tex]d=\frac{W}{x}[/tex]

so when substituting we get that:

[tex]d=\frac{2W}{W/6}[/tex]

when simplifying we get that:

d=12

Answer: 12 minutes

Step-by-step explanation:

This is a standard Work Formula question (2 or more 'entities' that work on a task together). When there are just 2 entities and there are no 'twists' to the question, we can use the Work Formula to get to the correct answer.

Work = (A)(B)/(A+B) where A and B are the individual times needed to complete the task.

We're told that two hoses take 20 minutes and 30 minutes, respectively, to fill a pool. We're asked how long it takes the two hoses, working together, to fill the pool.

(20)(30)/(20+30) = 600/50 = 12 minutes to fill the pool.

Answer:

There are no like terms.

Answer:

Step-by-step explanation:

$ 25 was the cost of each ​ticket before applying the service fee.

Step-by-step explanation:

Given that the cost of two tickets including the service fee = $59

So, the cost of one ticket including the service fee =  [tex]\frac{59}{2} = \$ 29.5[/tex]

Service fee on each ticket = 18%

Let the cost of 1 ticket without including the service fee = x

So, according to the data given in the question,

                     [tex]x+18 \% \text { of } x=29.5[/tex]

                     [tex]x+\frac{18}{100} \times x=29.5[/tex]

                    [tex]x+0.18 x=29.5[/tex]

                    [tex]1.18 x=29.5[/tex]

                    [tex]x=\frac{29.5}{1.18}=25[/tex]

Hence, cost of each ticket before applying the service fee = $25

Final answer:

The cost of each concert ticket before the 18% service fee was $25. This was calculated by dividing the total cost for two tickets with the service fee ($59) by the total percentage for two tickets (2.36).

Explanation:

The question asks to find the cost of each concert ticket before an 18% service fee is added, given that the total cost for two tickets including the service fee is $59.

To solve this problem, let's denote the cost of one ticket before the service fee as x. Since there is an 18% service fee per ticket, the total cost for one ticket including the service fee is x + 0.18x = 1.18x. As we have two tickets, their total cost would be 2 × 1.18x = 2.36x.

According to the given information, this total cost equals $59. So, we get the equation 2.36x = $59. To find the value of x, divide both sides of the equation by 2.36:

x = $59 ÷ 2.36 = $25

Therefore, the cost of each ticket before the service fee was $25.

Answer:

The total number of tennis ball did store sell in all is 1,356  

Step-by-step explanation:

Given as :

The number of sleeves of tennis ball in each box = 4

The number of tennis ball in each sleeve = 3

So, The total tennis ball in each box = 3 × 4 = 12

Now,

The selling of boxes on Saturday = 67

The selling of boxes on Sunday = 46

So, The Total number of boxes sold = 67 + 46 = 113

∵ The total tennis ball in each box = 12

∴ The total number of tennis ball did store sell in all = 113 × 12 = 1,356

Hence The total number of tennis ball did store sell in all is 1,356  Answer

Answer:

1356 Balls

Step-by-step explanation:

We first want to find out how many balls are in each box. There are [tex]$3\cdot4=\bold{12}$[/tex]balls per box. Then we find out how many boxes they sold. They sold [tex]$46+67=113$[/tex] boxes. Now we multiply.[tex]$113\cdot12=\bold{1356}$[/tex] balls/

Answer:

A. [tex]\frac{At}{60}=g[/tex]

Step-by-step explanation:

Consider the formula [tex]A = 60(\frac{g}{t})[/tex] is given,

We have to find : The formula for g.

For this we need to isolate g in one side of the equation,

Multiply both sides of the equation by t, ( using multiplicative property of equality )

We get,

[tex]At = 60g[/tex]

Divide both sides by 60 ( division property of equality )

We get,

[tex]\frac{At}{60}=g[/tex]

Which is the required equivalent equation.

That is,

OPTION A would be correct.

Answer: A. At/60 = g

Step-by-step explanation: Consider the formula  is given,

We have to find : The formula for g.

For this we need to isolate g in one side of the equation,

Multiply both sides of the equation by t, ( using multiplicative property of equality )

We get,

Divide both sides by 60 ( division property of equality )

We get,

Which is the required equivalent equation.

That is,

OPTION A would be correct.

Answer:

E

Step-by-step explanation:

For a function to be differentiable at a point, it must be continuous at that point [ f(x⁻) = f(x⁺) ], and smooth at that point [ f'(x⁻) = f'(x⁺) ].

f(-1⁻) = 3(-1) + 5 = 2

f(-1⁺) = -(-1)² + 3 = 2

So the function is continuous.

f'(-1⁻) = 3

f'(-1⁺) = -2(-1) = 2

So the function is not smooth.

Therefore, the derivative f'(-1) does not exist.

Answer:

Below in bold.

Step-by-step explanation:

Note that the lines q and m are parallel.

m < 1 = 180 - 110 = 70 degrees.  ( adjacent angles)

m < 2 =  180 - 115 = 65 degrees ( adjacent angles)

m < 3 = m < 2 = 65 degrees ( lines q and m are parallel and 2 and 3 are corresponding angles).

m < 4 = 180 - 65 = 115 degrees (adjacent to < 3).

m < 5 = 180 - m < 1 - m < 3 = 180 - 70 - 65 = 180 - 135 = 45 degrees. ( angles in a triangle add up to 180 degrees).

m < 6 = m < 5 = 45 degrees (  q and m are parallel - alternate angles).

m < 7 =- m < 6 = 45 degrees ( opposite angles).

Answer:

[tex]e^3[/tex]

Step-by-step explanation:

Given is a series as

[tex]1+\frac{3}{1!} +\frac{3^2}{2!} +...+\frac{3^n}{n!} +...[/tex]

Recall the expansion of

[tex]e^x = 1+x+\frac{x^2}{2!} +...+\frac{x^n}{n!} +...[/tex]

This expansion is valid for all real values of x.

Comparing this with our series we find that x =3

Hence the given series =[tex]e^3[/tex]

Thus we find that the given series can be recognized with the expansion of exponential series with powers of e and here we see that power of e is 3.

So the given Taylor series is equivalent to

[tex]e^3[/tex]

The series in the question is a Taylor series representing e^3x. At x=1, the sum of the series is exactly e^3.

The series you provided can be recognized as a Taylor series, an infinite sum of terms calculated from the values of a function's derivatives at a single point. Specifically, it resembles the Taylor series representation of the exponential function ex, which is 1 + x/1! + x2/2! + x3/3! + ... and so on.

Looking at your series 1 + 3/1! + 9/2! + 27/3! + 81/4! ..., we can see that each term 3n/n! is equivalent to (3n)/n!, which can be rewritten as (3^n)(1/n!). This yields a series in the form of 1 + 3x/1! + (3x)2/2! + (3x)3/3! + ... It's apparent that your series can be rewritten as e3x. So when x = 1, the sum of the series is e3, which is exactly the number e cubed.

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Answer:

Step-by-step explanation:

area[tex]=\frac{1}{2}*5*8\\=20~yd^2[/tex]

Answer:1.642 m

Step-by-step explanation:

Given

Initial Length of bridge [tex]L_0=3910 m[/tex]

temperature on cold day [tex]t_1=-5^{\circ}C[/tex]

temperature on hot day [tex]t_2=30^{\circ}C[/tex]

Change in temperature is [tex]\Delta T=30-(-5)=35^{\circ}C[/tex]

Coefficient of linear expansion of steel [tex]\alpha =12\times 10^{-6}/^{\circ}C[/tex]

and length after change in temperature is given by

[tex]L=L_0(1+\alpha \cdot \Delta T)[/tex]

[tex]\Delta =L_0\cdot \alpha \Delta T[/tex]

[tex]\Delta L=3910\cdot 12\times 10^{-6}\cdot 35[/tex]

[tex]\Delta L=1.642 m[/tex]

The Akashi Kaikyo Bridge in Japan will be longer during a warm summer day than on a cold winter day due to the thermal expansion of the steel. By calculating the linear thermal expansion using the formula ΔL = α * L * ΔT, with the coefficients for steel and the given temperature and length values, the change in length can be determined.

To understand how much longer the Akashi Kaikyo Bridge would be on a warm summer day compared to a cold winter day, we need to calculate the thermal expansion of the material that the bridge is made of, in this case, steel. The linear thermal expansion of a solid material can be calculated using the formula:

ΔL = α * L * ΔT,

where ΔL is the change in length of the material, α (alpha) is the coefficient of linear thermal expansion for the material, L is the original length of the material, and ΔT is the change in temperature.

For steel, the coefficient of linear thermal expansion α is typically about 12x10-6 1/°C. Given that the original length of the bridge, L, is 3910 m, and the temperature change, ΔT, is the difference between the summer and winter temperatures (30°C - (-5°C) = 35°C), we can substitute the given values into the formula to find ΔL. So:

ΔL = (12x10-6 1/°C) * (3910 m) * (35°C)

This will give you the change in length of the bridge between winter and summer. This thermal expansion across changing temperatures actually represents the bridge's natural ability to contract and expand without buckling, a key engineering aspect of all extended structures like bridges, roads and railways.

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150 different committees are possible

Given that a student dance committee is to be formed consisting of 2 boys and 4 girls

The membership is to be chosen from 5 boys and 6 girls

To find : number of different possible committees

A combination is a selection of all or part of a set of objects, without regard to the order in which objects are selected

The formula for combination is given as:

[tex]n C_{r}=\frac{n !}{(n-r) ! r !}[/tex]

where "n" represents the total number of items, and "r" represents the number of items being chosen at a time

We have to select 2 boys from 5 boys

So here n = 5 and r = 2

[tex]\begin{aligned} 5 C_{2} &=\frac{5 !}{(5-2) ! 2 !}=\frac{5 !}{3 ! 2 !} \\\\ 5 C_{2} &=\frac{5 \times 4 \times 3 \times 2 \times 1}{3 \times 2 \times 1 \times 2 \times 1} \\\\ 5 C_{2} &=10 \end{aligned}[/tex]

We have to select 4 girls from 6 girls

Here n = 6 and r = 4

[tex]\begin{aligned} 6 C_{4} &=\frac{6 !}{(6-4) ! 4 !}=\frac{6 !}{2 ! 4 !} \\\\ 6 C_{4} &=\frac{6 \times 5 \times 4 \times 3 \times 2 \times 1}{2 \times 1 \times 4 \times 3 \times 2 \times 1}=15 \end{aligned}[/tex]

Committee is to be formed consisting of 2 boys and 4 girls:

So we have to multiply [tex]5 C_{2}[/tex] and [tex]6 C_{4}[/tex]

[tex]5 C_{2} \times 6 C_{4}=10 \times 15=150[/tex]

So 150 different committees are possible

Final answer:

The question is a combinatorial problem in mathematics, where the goal is to calculate the number of different committees that can be formed from 5 boys and 6 girls. To solve this, the combinations formula is applied separately to choose 2 boys from 5 and 4 girls from 6, and the results are multiplied.

Explanation:

The question asks about the number of different committees that can be formed from a group of boys and girls. This is a combinatorial problem involving calculations to find the different possible combinations that can be made using a subset of a larger set. To solve this, you would use the combinations formula which is given by C(n, k) = n! / (k!(n-k)!), where n is the total number of items to choose from, k is the number of items to choose, and ! denotes the factorial of a number.

To determine how many different committees are possible, we calculate the number of ways to choose 2 boys from 5, and 4 girls from 6 separately, and then multiply these two results:

The number of ways to choose 2 boys from 5 is C(5, 2)

The number of ways to choose 4 girls from 6 is C(6, 4)

Therefore, the total number of different committees possible is C(5, 2) * C(6, 4).

Answer:

The solutions for 3 questions are explained one after the other below.

Step-by-step explanation:

1).Let x be the number of paperback books that she buys,  y be the number of hardback books that she buys.

for the first condition, i.e, she has decided to spend at most $150.00 on books,the required inequality will be :

[tex]8x+12y\leq 150[/tex]

for the second condition , i.e, she wants to purchase at least 12 books,

the required inequality will be:

[tex]x+y\geq 12[/tex]

2). the graph is in the attachment..

3). x,y are the two required solutions. where,

x =number of paperback books she buys.

y=number of hardback books she buys.

Answer:

1) equations 1, 2, 3  and 4

2)  see picture attached (the region of the solutions is in yellow)

3) x = 18.75 and y =0

   y = 12.5 and x =0

Step-by-step explanation:

Let's call x the number of paperback books bought and y the number of  hardback books bought.

She decided to spend at most $150.00. All paperback books are $8.00 and all hardback books are $12.00. Combining this information we get:

x*8 + y*12 ≤ 150 (eq.  1)

Anna wants to purchase at least 12 books. Mathematically:

x + y ≥ 12 (eq. 2)

On the other hand,  both the number of paperback books bought and the number of  hardback books bought must be positive, that is:

x ≥ 0 (eq. 3)

y ≥ 0 (eq. 4)

3) One possible solution is got if we make y = 0 and to take eq. 1 as an equality, then:

From eq. 1: x*8 = 150

x = 150/8 = 18.75

equations 2 and 3 are also satisfied

Another option is to make x = 0 and to take eq. 1 as an equality, then:

From eq. 1: y*12 = 150

y = 150/12 = 12.5

equations 2 and 4 are also satisfied

Answer:

Yes, those are the first triangular numbers.

There is a relation between the number and its position but isn't direct.

Step-by-step explanation:

The triangular numbers can be represented by equilateral triangles, but also can be represented by:

[tex]T_{n} = \frac{n(n+1)}{2}[/tex]

where:

n, represents the position

T represent the triangular number.

As you may see, the equation of triangular numbers is not a straight line. It is a parable. For that reason there isn't a direct variation.

Answer:

No, the triangular numbers are not a direct variation. There is not a constant of variation between a number and its position in the sequence. The ratios of the numbers to their positions are not equal. Also, the points (1, 1), (2, 3), (3, 6), and so on, do not lie on a line.

Step-by-step explanation:

Answer: the amount of money raised by the three for charity in all is 386.81

Step-by-step explanation:

Alexis raises 75.23 for charity.

Sue raises 3 times as much as Alexis. This means that the total amount of money raised by Sue is 3 × 75.23= 225.69

The amount of money that Manuel raises is 85.89.

The amount of money raised by all of them(Alexis, Sue and Manuel) would be sum of the amount of money raised by Alexis + the amount of money raised by Sue +

the amount of money raised by Manuel. This becomes

75.23 + 225.69 + 85.89 = 386.81

Answer: 386.81

Step-by-step explanation:

Alexia raises 75.23 = x

Sue raises 3 times Alexia = 3x

= 3 × 75.23

=225.69

Manuel raises 85.89

Total amount raised = Alexia + sue + Manuel

= 75.23 + 225.69 + 85.89

= 386.81

The person that traveled the most distance is Julie.

An expression is a way of writing a statement with more than two variables or numbers with operations such as addition, subtraction, multiplication, and division.

Example: 2 + 3x + 4y = 7 is an expression.

We have,

From the figure,

Julie:

Total distance covered.

= sports complex to school + school to mall + mall + library

= 2/3 + 2/5 + 1(1/3)

= 2/3 + 2/5 + 4/3

= (10 + 6 + 20)/15

= 36/15

= 12/5

= 2(2/5) miles

= 2.4 miles

Kyle:

Total distance covered.

= house to mall + mall to library

= 4/5 + 1(1/3)

= 4/5 + 4/3

= (12 + 20)/15

= 32/15

= 2(2/15) miles

= 2.13 miles

Thus,

Julie has traveled more distance than Kyle.

Learn more about expressions here:

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Answer:

The length of the field = 24.5 yards

The width of the field = 12 yards.

Step-by-step explanation:

If "w" is the width, then the length is 49 - 2w.

The area of the rectangle field = length × width

= w(49 - 2w)

Area = [tex]49w - 2w^2[/tex]

This a quadratic equation, the vertex of x coordinate is w

w = [tex]\frac{-b}{2a}[/tex]

Here a = -2 and b = 49

w = [tex]\frac{-49}{2(-2)} = \frac{-49}{-4} = 12.25[/tex].

So width of the field is 12.25 yards.

The length of the filed = 49 - 2(12.25) = 24.5

Area = 12.25 × 24.5 = 300.125 square yards.

The field has an area of 294.

Therefore, the length must be 24.5 yards and width must be 12 yards.

24.5 × 12 =294 square yards.

Therefore, the length of the field = 24.5 yards

the width of the field = 12 yards.

To find the dimensions of the field, we can use equations based on the area and perimeter. We set up equations for the area and perimeter and solve them simultaneously to find the values of width and length.

To find the dimensions of the field, we can use the formula for finding the area of a rectangle: length x width. Let's assume the length of the field is x yards. Since the width is the smaller dimension and requires two sides, we'll represent it as 2w yards. Given that the area of the field is 294 sq-yards, we have the equation x * 2w = 294.

Additionally, we know that the farmer has 49 yards of fence, and he needs to enclose three sides of the field. The three sides are two sides of width (2w) and one side of length (x). So, the total length of the fence needed is 2w + 2w + x = 49.

Simplifying the equation, we have 4w + x = 49. Substituting x from the first equation into the second equation, we get 4w + (294 / 2w) = 49.

Now, we can solve this equation to find the value of w. Once we have the value of w, we can substitute it back into the first equation to find the value of x.

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Answer:

Range is 57.

Variance is 375.143.

standard deviation is 19.37.

Step-by-step explanation:

Consider the provided information.

Range is the difference between highest and lowest data value.

The highest data value is 57 and lowest is 0.

Thus the range is 57-0=57

Range is 57.

Mean is the sum of data value divided by the number of data value:

[tex]\bar x=\frac{31+52+35+57+0+32+35+52+46+41+30+41+0+0}{14}\approx32.286[/tex]

The variance is the sum of squared deviation from the mean divided by n-1.

[tex]s^2 =\frac{\sum(x_i -\bar x)^2}{n - 1}[/tex]

Substitute the respective values in the above formula we get:

[tex]s^2=\frac{(31 - 32.286)^2 +(52-32.286)^2+ ... + (0 -32.286)^2}{14 - 1}\approx 375.143[/tex]

Hence, the variance is 375.143.

Standard deviation is square root of variance.

standard deviation = [tex]\sqrt{375.143}[/tex]

standard deviation ≈ 19.37

Hence, standard deviation is 19.37.

For all cans consumed, the statistics are not representative of the population because in the calculations each brand is weighted equally. Each of the 14 brands of soda is unlikely to be consumed in the same way.

It is very unlikely that all 14 drinks are consumed equally. So,given data is not representative of population

The football stadium was approximately  5,000  thousand square feet.

The given cost function is [tex]\( C(x) = \frac{7,250,000}{x} + 60 \)[/tex], where C(x)  represents the cost to build a football stadium of  x  thousand square feet. To find the size of the stadium when the cost is $8,000, we set C(x)  equal to $8,000 and solve for  x :

[tex]\[ 8,000 = \frac{7,250,000}{x} + 60 \][/tex]

Subtracting 60 from both sides:

[tex]\[ 7,940 = \frac{7,250,000}{x} \][/tex]

Now, solving for  x , we multiply both sides by  x :

[tex]\[ x = \frac{7,250,000}{7,940} \][/tex]

Performing the division:

x approx 912.66

Since  x  represents the size of the stadium in thousand square feet, the final answer is approximately 912.66  thousand square feet. Therefore, the stadium was approximately  5,000  thousand square feet.

In summary, to determine the size of the football stadium, we set the cost function equal to the given cost, solved for  x , and found that the stadium was approximately  912.66  thousand square feet.

The football stadium was approximately x = 916.67 thousand square feet.

The given cost function for building a football stadium is [tex]\(C(x) = \frac{7,250,000}{x} + 60\)[/tex], where \(x\) represents the size of the stadium in thousand square feet. We are asked to find the size of the stadiumx when the cost[tex](\(C(x)\))[/tex] is $8,000.

To solve for x, we set [tex]\(C(x)\)[/tex] equal to $8,000 and solve for x:

[tex]\[ 8,000 = \frac{7,250,000}{x} + 60 \][/tex]

First, subtract 60 from both sides:

[tex]\[ 7,940 = \frac{7,250,000}{x} \][/tex]

Next, multiply both sides by x to isolate x in the denominator:

[tex]\[ 7,940x = 7,250,000 \][/tex]

Finally, solve for [tex]\(x\)[/tex] by dividing both sides by 7,940:

[tex]\[ x = \frac{7,250,000}{7,940} \approx 916.67 \text{ thousand square feet} \][/tex]

Therefore, the football stadium was approximately 916.67 thousand square feet in size.

In summary, by substituting the given cost into the cost function and solving the resulting equation, we find that the stadium size is approximately 916.67 thousand square feet. This process involves algebraic manipulation to isolate [tex]\(x\)[/tex] and perform the necessary arithmetic calculations.e

Answer:

Step-by-step explanation:

The equation of a quadratic in vertex form is ...

  y = a(x -h)² +k

The coordinates of the vertex are (h, k), which means the axis of symmetry is x=h. All of your equations have h=5, so their axis of symmetry is x = 5.

__

For a > 0, the parabola opens upward; for a < 0, it opens downward. The first three equations have a > 0, so open upward. The last one opens downward.

A quadratic equation is a second-order polynomial with real solutions represented by the quadratic formula. When graphically represented, these equations produce a curved line, and in the context of physical data, only positive roots often matter. Furthermore, vectors can form a right angle triangle with their components.

The question pertains to understanding quadratic equations and their properties. In mathematical terms, a quadratic equation is a second-order polynomial equation in a single variable with a form of ax² + bx + c = 0, where x represents an unknown, and a, b, and c are constants. Note though a ≠ 0. If a =0, then the equation is linear, not quadratic. The constants a, b, and c are referred to as the coefficients of the equation.

The solutions of these quadratic equations are given by the quadratic formula: -b ± √(b² - 4ac) / 2a.

Furthermore, when plotting the relationship between any two properties of a system which can be represented through a quadratic equation, the graph is a two-dimensional plot with a curve, indicative of the quadratic relationship. Specifically for physical data, quadratic equations always have real roots often only positive values hold significance.

Lastly, it's true that a vector can form the shape of a right angle triangle with its x and y components. This statement doesn't directly involve a quadratic equation but still ties into the broader umbrella of mathematical functions.

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Answer:

(a). 15

(b). 78

Step-by-step explanation:

Growth of the population of a fruit fly is modeled by

N(t) = [tex]\frac{600}{1+39e^{-0.16t} }[/tex]

where t = number of days from the beginning of the experiment.

(a). For t = 0 [Initial population]

N(0) = [tex]\frac{600}{1+39e^{-0.16\times 0} }[/tex]

       = [tex]\frac{600}{1+39}[/tex]

       = [tex]\frac{600}{40}[/tex]

       = 15

Initial population of the fruit flies were 15.

(b).Population of the fruit fly colony on 11th day.

N(11) = [tex]\frac{600}{1+39e^{-0.16\times 11} }[/tex]

       = [tex]\frac{600}{1+39e^{-1.76} }[/tex]

       = [tex]\frac{600}{1+39\times 0.172 }[/tex]

       = [tex]\frac{600}{1+6.71}[/tex]

       = [tex]\frac{600}{7.71}[/tex]

       = 77.82

       ≈ 78

On 11th day number of fruit flies colony were 78.

Answer:

  a.)  [1, infinity)

Step-by-step explanation:

The equation is that of a parabola that opens upward with vertex (1, 1). Hence the minimum value of f(x) is 1, and all values greater than that are part of the range: [1, ∞).

_____

The "vertex form" of the equation of a parabola is ...

  f(x) = a(x -h)^2 + k

The vertex is at (h, k). When a > 0, the parabola opens upward. When a < 0, the parabola opens downward. Whichever way it opens, the value k is an extreme value and the limit of the range.

The measure of PQ is 12.72

We have the right triangle ΔPQR and we want to know the measure of PQ. PQ is opposite to ∠R, so from trigonometry we know that:

[tex]sin(\alpha)=\frac{Opposite \ side}{Hypotenuse} \\ \\ \\ Here: \\ \\ \alpha=m\angle R=58^{\circ} \\ \\ Opposite \ side=\overline{PQ} \\ \\ Hypotenuse=\overline{RQ}=15 \\ \\ \\ So: \\ \\ sin(58^{\circ})=\frac{\overline{PQ}}{15} \\ \\ \\ Isolating \ \overline{PQ}: \\ \\ \overline{PQ}=15sin(58^{\circ}) \\ \\ \overline{PQ}=15(0.848)\\ \\ \boxed{\overline{PQ}=12.72}[/tex]

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Answer:

Step-by-step explanation:

Answer:

Area of Δ ABC = 21.86 units square

Perimeter of Δ ABC = 24.59 units

Step-by-step explanation:

Given:

In Δ ABC

∠A=45°

∠C=30°

Height of triangle = 4 units.

To find area and perimeter of triangle we need to find the sides of the triangle.

Naming the end point of altitude as 'O'

Given [tex]BO\perp AC[/tex]

For Δ ABO

Since its a right triangle with one angle 45°, it means it is a special 45-45-90 triangle.

The sides of 45-45-90 triangle is given as:

We are given BO (Leg 1) [tex]x=4[/tex]

∴ AO (Leg2) [tex]=x=4[/tex]

∴ AB (hypotenuse) [tex]=x\sqrt2=4\sqrt2=5.66 [/tex]  

For Δ CBO

Since its a right triangle with one angle 30°, it means it is a special 30-60-90 triangle.

The sides of 30-60-90 triangle is given as:

We are given BO (side opposite 30° angle) [tex]=x=4[/tex]

CO (side opposite 60° angle) [tex]=x\sqrt3=4\sqrt3=6.93[/tex]

BC (Hypotenuse) [tex]=2x=2\times 4 =8[/tex]

Length of side AC is given as sum of AO and CO

[tex]AC=AO+CO=4+6.93=10.93[/tex]

Perimeter of Δ ABC= Sum of sides of triangle

⇒ AB+BC+AC

⇒ [tex]5.66+8+10.93[/tex]

⇒ [tex]24.59[/tex] units

Area of Δ ABC = [tex]\frac{1}{2}\times base\times height[/tex]

⇒  [tex]\frac{1}{2}\times 10.93\times 4[/tex]

⇒ [tex]21.86[/tex] units square

Answer: The angle of depression is 18.3 degrees

Step-by-step explanation:

The scenario is illustrated in the attached photo

A right angle triangle ABC is formed. The angle of depression, theta forms an alternate angle of # on the ground. To determine the angle, #, we would apply trigonometric ratio

Tan# = opposite side / adjacent side

Opposite side = BC = 10 feets

Adjacent side = AB = 30 feets

Tan # = 10/30 = 0.33

# = tan^-1(0.33)

# = 18.2629

Approximately 18.3 degrees

Since # is alternate to the angle of depression, it means that they are equal. So

The angle of depression is 18.3 degrees

Final answer:

The number of different color groupings the rug manufacturer can use is found by calculating combinations of 7 colors taken 5 at a time, which is 42 different color combinations.

Explanation:

The question asks for the number of different color groupings using seven colors taken five at a time without repetitions. This is a problem of combinations, which is a part of mathematics. To find the number of combinations, we use the formula for combinations which is:
C(n, k) = n! / (k! * (n - k)!)

where n is the total number of items, k is the number of items to choose, n! is the factorial of n, and k! is the factorial of k. Applying this to the given problem:

n = 7 (since there are seven colors)

k = 5 (since we are selecting five colors for each rug)

Therefore, the number of ways to select 5 colors from 7 is:
C(7, 5) = 7! / (5! * (7 - 5)!) = 7! / (5! * 2!) = (7*6) / (2*1) = 42

Thus, there are 42 different color combinations that the manufacturer can advertise for the rugs.

There are 120 ways for the five individuals to arrive at the dinner party. If Eduardo always arrives first and Tyrone always arrives last, there are 6 possible orderings. The chance of this specific circumstance happening is 5%.

The subject of this question is Permutations and Probability in mathematics.

a. In how many ways can they arrive?

Since there are 5 people and each can arrive at different times, the number of ways they can arrive is equal to the number of permutations of 5 distinct items. This can be calculated as 5 factorial (5!) which equals 5 * 4 * 3 * 2 * 1 = 120. Thus, there are 120 different ways they can arrive.

b. In how many ways can Eduardo arrive first and Tyrone last?

If Eduardo arrives first and Tyrone arrives last, this means there are 3 people left (Sarah, Maria, and Jim) who can arrive in any order in the middle. The number of permutations for these 3 is 3 factorial (3!) which equals 3 * 2 * 1 = 6. Thus, there are 6 different ways Eduardo can arrive first and Tyrone last.

c. Find the probability that Eduardo will arrive first and Tyrone will arrive last.

Probability is calculated by dividing the number of favorable outcomes by the total number of outcomes. From part b, we know that there are 6 favorable outcomes (Eduardo first, Tyrone last). From part a, we know there are 120 total outcomes. Thus, the probability is 6 / 120 = 0.05 or 5%.

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Answer: $15966.08

Step-by-step explanation:

The formula to calculate the compound amount is given by :-

[tex]A=P(1+r)^t[/tex]

, where P = initial deposit amount.

Time = Time period .

r= Rate of interest in decimal  (compounded once per year)

Given : P= $  12000

r= 7.4 percent =0.074

t= 4 years

Then, the compound amount after 4 years :

[tex]A=12000(1+0.074)^{4}\\\\=12000(1.074)^4=12000(1.33050688258)\\\\=15966.082591\approx15966.08[/tex]

Hence, compound amount after 4 years = $15966.08

After four years, Frank will owe approximately $16,140.

To determine the amount Frank will owe after four years, we can use the formula for compound interest:

[tex]\[ A = P(1 + \frac{r}{n})^{nt} \][/tex]

Given:

-  P = $12,000  (the initial amount borrowed)

-  r = 7.4% = 0.074 (annual interest rate)

-  n = 1  (compounded once per year)

-  t = 4  years (time period in question)

Plugging these values into the compound interest formula, we get:

[tex]\[ A = \$12,000(1 + \frac{0.074}{1})^{1 \times 4} \] \[ A = \$12,000(1 + 0.074)^{4} \] \[ A = \$12,000(1.074)^{4} \][/tex]

Now, we calculate [tex]\( (1.074)^{4} \)[/tex]:

[tex]\[ A = \$12,000 \times 1.345 \] \[ A \approx \$16,140 \][/tex]

Finally, we multiply this by the principal amount to find out how much will be owed after four years:

[tex]\[ A = \$12,000 \times 1.345 \] \[ A \approx \$16,140 \][/tex]