Chance has hired a construction crew to renovate his kitchen. They charge $3.89 per square foot for materials and $121.26 per day of labor. Chance spent $2,982.68 on the renovation. If the number of square feet is 252 more than the number of days it took for the renovation, how long did the renovation take?
A.
19 days
B.
16 days
C.
3 days
D.
14 days

Answer:

  14 feet

Step-by-step explanation:

There are 3 feet in 1 yard, so 36 feet in 12 yards. The remaining ribbon will be the original amount less the amount used.

  36 - 22 = 14 . . . . feet remaining

After decorating the blankets, Olga will have 14 feet of ribbon remaining. The conversion from yards to feet and subtraction calculates this remaining amount accurately.

Calculating Remaining Ribbon

To determine how much ribbon Olga has left after decorating the blankets, we need to perform a couple of conversions and a subtraction.

First, let's convert the total ribbon from yards to feet:

→ 1 yard = 3 feet

→ 12 yards = 12 * 3

                 = 36 feet

Next, Olga uses 22 feet of ribbon to decorate the blankets:

→ Total ribbon in feet: 36 feet

→ Ribbon used: 22 feet

Now, subtract the amount used from the total:

→ Remaining ribbon = 36 feet - 22 feet

                                 = 14 feet

Olga will have 14 feet of ribbon remaining.

Answer:

[tex]m(d)=26\ d[/tex]  

where [tex]m[/tex] represents miles driven

and [tex]d[/tex] represents number of days driven.

Step-by-step explanation:

Given:

Total distance driven in miles = 70,000

Average distance driven each day = 26 miles

Taking average as unit rate of miles covered per day.

∴ we can say the car covers 26 miles per day.

Using unitary method to find miles driven in [tex]d[/tex] days.

In 1 day miles driven = 26

In [tex]d[/tex] days miles driven = [tex]26\times d =26\ d[/tex]

So, to find [tex]m[/tex] miles driven the expression can be written as:

[tex]m(d)=26\ d[/tex]  

Answer:

Part 1) An expression for the x-coordinate of T is (a+2)

Part 2) The value of x=39 ft (see the explanation)

Step-by-step explanation:

Part 1)

step 1

we know that

The rule of the reflection of a point across the x-axis is equal to

[tex](x,y) -----> (x,-y)[/tex]

Apply the rule of the reflection across the x-axis to the Q coordinates

Q (a,b) ----------> Q'(a,-b)

step 2

The translation is 2 units to the right

so

The rule of the translation is

(x,y) ----> (x+2,y)

Apply the rule of the translation to the Q' coordinates

Q'(a,-b) -----> T(a+2,-b)

therefore

An expression for the x-coordinate of T is (a+2)

Part 2)

we know that

A reflection is a rigid transformation, the image is the same size and shape as the pre-image

In this problem the floor plan house A and the floor house B have the same size and shape

That means that its corresponding sides and corresponding angles are congruent

therefore

The value of x=39 ft

Answer:

Step-by-step explanation:

In 9 minutes it would make 9/45 = 1/5 th of a revolution.

360(1/5) = 72 degrees

Coordinates:

(30cos72, +/- 30sin72) [+ for counterclockwise, - for clockwise)

(9.3ft, +/- 28.5ft)

https://answers.yahoo.com/question/index?qid=20100518113431AAvVZGM&guccounter=1&guce_referrer=aHR0cHM6Ly93d3cuZ29vZ2xlLmNvbS8&guce_referrer_sig=AQAAADh3so85ELjMHYh-Bs4EBfWYXngnX6lb_DUJTEV63qEAk8V25pxPyGr-fVlMatpwaXP-ke4TzkPe-OvBlV2WAwZ7iw3e4lBodLi1bt5txsrb6ccIGmS2tQHRymY0UsEqHUDfdNuDJKZRsnZ_gUPM1ChwoRTmeHDtRf2vrisw_B_e

The coordinates of the table after 9 minutes are approximately (9.3, 28.5).

Coordinates are a collection of numbers that aid in displaying a point's precise location on the coordinate plane.

Since the restaurant takes 45 minutes to complete a full counter clockwise rotation, its angular velocity is:

ω = (2π radians) / (45 minutes)

≈ 0.1396 radians per minute

If we let θ be the angle between the position of the table and the positive x-axis at time t, then the position of the table can be expressed as:

x = 30 cos(θ)

y = 30 sin(θ)

To find the position of the table after 9 minutes, we can use the angular velocity to determine the angle that the restaurant has rotated. After 9 minutes, the angle of rotation is:

θ = ωt = 0.1396 radians/minute x 9 minutes

≈ 1.256 radians

Using the values of θ and the radius of 30 ft, we can find the coordinates of the table:

x = 30 cos(1.256) ≈ 9.3

y = 30 sin(1.256) ≈ 28.5

Learn more about Coordinates here:

https://brainly.com/question/16634867

#SPJ3

Answer: BD = 108

Step-by-step explanation:

In a parallelogram, the opposite sides are congruent and the diagonals bisect each other. It means that they bisect at a midpoint that divides them equally.

Therefore,

AB = DC

AD = BC

BD = AC

Also BE = ED. This means that

7x - 2 = x^2 - 10

x^2 - 10 +2 - 7x = 0

x^2 - 7x -8 = 0

Solving the quadratic equation with factorization method,

x^2 + x - 8x -8 = 0

x(x + 1) -8(x + 1) = 0

x - 8 = 0 or x + 1 = 0

x = 8 or x = -1

Since x cannot be negative,

x = 8

BE = 7×8 - 2 = 54

ED = 8^2 - 10 = 54

BD = BE + ED = 54 +54 = 108

Answer:Shanny and her friends wanted to raise $2400

Step-by-step explanation:

Fundraising events were held by Shanika and her friends to help pay for repairs.

Let x = the total amount of money that Shanika and her friends want to raise during the fund raising events. After the first event, they raise $240,which is 10% of the total amount that they want to raise. This means that after the first event, they raised 10/100 ×x = 0.1x

This 0.1x that they raised is equal to $240. Therefore,

0.1x = 240

x = 240/0.1 = 2400

Shanny and her friends wanted to raise $2400

Answer:

$2,400 is the correct answer

Step-by-step explanation:

Answer: There are 2635 acres covered 24 hours later.

Step-by-step explanation:

Since we have given that

At time t = 0, number of acres forest fire covers = 2008 acres

We first consider the equation:

[tex]A=\int\limits^t_0 {8\sqrt{t}} \, dt\\\\A=8\dfrac{t^{\frac{3}{2}}}{\frac{3}{2}}+C\\\\A=\dfrac{16}{3}t^{\frac{3}{2}}+C[/tex]

At t=0, A= 2008

So, it becomes,

[tex]2008=C[/tex]

So, now it becomes,

[tex]A=\dfrac{16}{3}t^{\frac{3}{2}}+2008\\\\At\ t=24,\\\\A=\dfrac{16}{3}(24)^{\frac{3}{2}}+2008\\\\A=2635.06[/tex]

Hence, there are 2635 acres covered 24 hours later.

Step-by-step explanation:

Let w be the work of stock the shelves and t be the time for Camille to the worl alone.

Dolores takes 7.2 hours.

[tex]\texttt{Rate of Dolores = }\frac{w}{7.2}[/tex]

[tex]\texttt{Rate of Camille = }\frac{w}{t}[/tex]

If they combine work is completed in 2.8 hours.

          That is

                       [tex]2.8=\frac{w}{\frac{w}{7.2}+\frac{w}{t}}\\\\2.8=\frac{7.2t}{t+7.2}\\\\2.8t+20.16=7.2t\\\\4.4t=20.16\\\\t=4.58hours[/tex]

   It takes 4.58 hours to stock the shelves if Camille were working alone

Answer:

d=1.416.525 mile

Step-by-step explanation:

V1=5151m/h, t1=t, V2=7575m/h, t2=t-88h

d1=d2 Because it is same distance; V1=d1/t and V2=d2/(t-88) but d1=d2

d=V1t=V2(t-88) → 5151t=7575(t-88) → 5151t=7575t-666.600 → 7575t-5151t=666.600 → 2424t=666.600 → t=666.600/2424 → t=275h so

[tex]d=5151\frac{mile}{h}.275h =  1.416.525mile[/tex]

The formula for the value of nth term is [tex]a_{n}[/tex] = 3n + 1

Step-by-step explanation:

The formula of the nth term in the arithmetic sequence is

[tex]a_{n}=a+(n-1)d[/tex] , where

∵ The first term of an arithmetic sequence is equal to four

∴ a = 4

∵ The common difference is equal to three

∴ d = 3

- Substitute these values in the rule of the nth term

∵ [tex]a_{n}=a+(n-1)d[/tex]

∴ [tex]a_{n}=4+(n-1)3[/tex]

- Simplify it

∴ [tex]a_{n}=4+3n-3[/tex]

∴ [tex]a_{n}=1+3n[/tex]

The formula for the value of nth term is [tex]a_{n}[/tex] = 3n + 1

Learn more:

You can learn more about the sequence in brainly.com/question/7221312

#LearnwithBrainly

The right answer is Option D.

Step-by-step explanation:

Given,

Total people surveyed = 250

Total people who prefer sports channel= 62

Percent of people who prefer sports channel;

Percent = [tex]\frac{people\ who\ prefer\ sports\ channel}{Total\ no.\ of\ people\ surveyed}*100[/tex]

[tex]Percent=\frac{62}{250}*100\\\\Percent=\frac{6200}{250}\\\\Percent= 24.8\%[/tex]

24.8% of everyone surveyed preferred sports channel.

The right answer is Option D.

Keywords: percentage, division

Learn more about percentages at:

#LearnwithBrainly

Answer:

61°C

Step-by-step explanation:

Newton's Law  of cooling gives the temperature -time relationship has:

T (t)  = 20 + 75 е⁻(t/50)-------------------------------------------------------- (1)

where Time  is in  minutes (min) & Temperature in degree Celsius (°C)

During the first half hour, t = 30 mins

Substituting into (1)

T = 20 + 75  е⁻(30/50)

  = 20 + 75(0.5488)

  =  20 + 41.16

  =  61.16°C

  ≈    61°C

Answer:

You didn't write the expression

Step-by-step explanation:

Answer:

Step-by-step explanation:

Motrcycle: 2 hours  150/75 = 2 hours

Bus:  150/60= 2.5 hours

Truck:150/50= 3 hours

Bike:= 150/20= 7.5 hours

Answer:

Step-by-step explanation:

what?

Answer:

Below.

Step-by-step explanation:

The x and y coordinates flip with this reflection e, g   (2, 1) goes to (1, 2).

So  R (0, -5) ---> R' (-5, 0).

     E (4, -3) ---> E' (-3, 4).

     C (6, -5) ---> C' (-5, 6).

    T (2, -7) ---> T' (-7, 2).

For this case we have the following equation:

[tex]I = \frac {nE} {nr + R}[/tex]

We must clear the variable "n", for them we follow the steps below:

We multiply by [tex]nr + R[/tex] on both sides of the equation:

[tex]I (nr + R) = nE[/tex]

We apply distributive property on the left side of the equation:

[tex]Inr + IR = nE[/tex]

Subtracting [tex]nE[/tex] from both sides of the equation:

[tex]Inr-nE + IR = 0[/tex]

Subtracting IR from both sides of the equation:

[tex]Inr-nE = -IR[/tex]

We take common factor n from the left side of the equation:

[tex]n (Ir-E) = - IR[/tex]

We divide between Ir-E on both sides of the equation:

[tex]n = - \frac {IR} {Ir-E}[/tex]

Answer:

[tex]n = - \frac {IR} {Ir-E}[/tex]

Answer: 0.0047

Step-by-step explanation:

Given : A manufacturer knows that their items have a normally distributed length, with a mean of 7.1 inches, and standard deviation of 1.7 inches.

i.e. [tex]\mu=7.1\text{ inches}[/tex]

[tex]\sigma=17\text{ inches}[/tex]

Sample size : n= 24

Let [tex]\overline{X}[/tex] be the sample mean.

Formula : [tex]z=\dfrac{\overline{x}-\mu}{\dfrac{\sigma}{\sqrt{n}}}[/tex]

Then, the probability that their mean length is less than 6.2 inches will be :-

[tex]P(\overline{x}<6.2)=P(\dfrac{\overline{x}-\mu}{\dfrac{\sigma}{\sqrt{n}}}<\dfrac{6.2-7.1}{\dfrac{1.7}{\sqrt{24}}})\\\\\approx P(z<-2.6)\\\\=1-P(z<2.6)\ \ [\because\ P(Z<-z)=1-P(Z<z)]\\\\=1-0.9953=0.0047\ \ \ [ \text{Using z-value table}][/tex]

hence,. the required probability = 0.0047

Answer:

0.0047

step-by-step explanation

The magnitude of the angular momentum of the particle relative to the point (0.9 m, 10 m) is [tex]{\text} 32.8 kg m^2/s[/tex].

Angular momentum is a physical quantity that measures the rotational motion of an object or system.

Given:

Mass = 4 kg,

velocity = 2 m/s

The following equation provides the particle's angular momentum (L):

L = mvr

where:

m = mass

v = velocity of the particle

r = perpendicular distance

To find the magnitude of the angular momentum relative to the point point [tex](x_0, y_0)[/tex], where [tex]x_0[/tex] = 0.9 m and [tex]y_0[/tex] = 10 m.

To find the perpendicular distance (r), use the distance formula:

[tex]r = \sqrt{((x - x0)^2 + (y - y0)^2)[/tex]

Substituting the values [tex]x_0[/tex] = 0.9 m and [tex]y_0[/tex] = 10 m in above formula

[tex]r = \sqrt{((0 - 0.9)^2 + (6-10)^2)[/tex]

 = [tex]\sqrt{((-0.9)^2 + (-4)^2)[/tex]

 = √(0.81 + 16)

 = √16.81

 = 4.1 m

Now, the angular momentum (L) using the formula:

L = mvr

L = 4 kg x 2 m/s x 4.1 m

L = 32.8 kg

As a result, the particle's angular momentum is  [tex]{\text} 32.8 kg m^2/s[/tex].

Learn more about Angular momentum here:

https://brainly.com/question/33408478

#SPJ12

The magnitude of the angular momentum relative to a point (x0, y0) depends on the moment of inertia and angular velocity of the particle. However, in this case, the angular velocity is undefined, so the magnitude of the angular momentum is also undefined.

The angular momentum of a particle can be calculated by multiplying its moment of inertia with its angular velocity. In this case, the particle has a mass of 4 kg and moves with a constant speed of 2 m/s in the positive x-direction. To find the magnitude of its angular momentum relative to the point (x0, y0), we need to calculate the moment of inertia and angular velocity. Since the particle is moving in the xy-plane, we can calculate the distance of the particle from the point (x0, y0) and use it to find the angular momentum. The magnitude of the angular momentum can be calculated by dividing the cross product of the position vector and linear momentum with the mass of the particle.

First, let's calculate the moment of inertia (I) of the particle. The moment of inertia can be calculated using the formula I = mr², where m is the mass of the particle and r is the distance of the particle from the axis of rotation. In this case, the particle is moving in the xy-plane, so the distance of the particle from the point (x0, y0) can be calculated using the distance formula: d = sqrt((x-x0)² + (y-y0)²). Substituting the values, we have d = sqrt((0-0.9)² + (6-10)²) = sqrt(13.21) = 3.63 m. The moment of inertia can be calculated as I = 4 kg * (3.63 m)² = 52.60 kg*m².

Next, let's calculate the angular velocity (ω) of the particle. The angular velocity can be calculated using the formula ω = v/r, where v is the linear velocity of the particle and r is the distance of the particle from the axis of rotation. In this case, the particle has a constant speed of 2 m/s in the positive x-direction along y = 6 m, so the distance of the particle from the axis of rotation is the distance from the point (0, 6). Substituting the values, we have r = sqrt((0-0)² + (6-6)²) = sqrt(0) = 0 m. The angular velocity can be calculated as ω = 2 m/s / 0 m = undefined. Since the angular velocity is undefined, the magnitude of the angular momentum relative to the point (x0, y0) is also undefined.

https://brainly.com/question/37906446

#SPJ11

Answer:

(3) ∠BCA ≅ ∠DAC

Step-by-step explanation:

BC and AD are parallel.  AC is a transversal line passing through both lines.  That means ∠BCA and ∠DAC are alternate interior angles.  Therefore, they are congruent.

Answer:

The answer is $1000.

Step-by-step explanation:

The policy was issued with a $500 deductible and a limit of four deductibles per calendar year.

As given that two claims were paid in September 2013, each incurring medical expenses in excess of the deductible.

So, the family's out-of-pocket medical expenses for 2013 will be :

[tex]500+500=1000[/tex] dollars

As the limit was up to 4 deductibles in a calendar year, and in 2013, there were 2 claims, so that sums up to be $1000.

The family's out-of-pocket medical expenses for 2013 would be $1000, as they paid the $500 deductible for each of the two claims made that year, with their health insurance policy limiting to four deductibles per year.

The subject of the question involves calculating the out-of-pocket medical expenses for a family under their health insurance policy, which includes understanding how deductibles work. In the scenario given, the family purchased a policy with a $500 deductible and a limit of four deductibles per calendar year. In 2013, they made two claims where each exceeded the deductible amount. Therefore, their out-of-pocket expenses for 2013 would be two times the deductible amount, since the policy has a limit of four deductibles per year but only two claims were filed and paid within that year.

Mathematically, this can be calculated as:

Total out-of-pocket expenses for 2013: $500 + $500 = $1000.

https://brainly.com/question/36307018

#SPJ6

It can be done in 50388 ways

Step-by-step explanation:

When the selection has to be made without order, combinations are used.

The formula for combination is:

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

Here

Total books = n =19

Books to be chosen = r = 7

Putting the values

[tex]C(19,7) = \frac{19!}{7!(19-7)!}\\\\=\frac{19!}{7!12!}\\\\=50388\ ways[/tex]

It can be done in 50388 ways

Keywords: Combination, selection

Learn more about combinations at:

#LearnwithBrainly

Answer:345

Step-by-step explanation:

Final answer:

To construct a 99% confidence interval for the calorie content of the energy bars, calculate the standard error, find the appropriate t-value, then compute the margin of error, and add and subtract it from the sample mean. The resulting 99% CI for the true mean calorie content is approximately (214.59, 245.41) calories.

Explanation:

To construct a 99% confidence interval (CI) for the true mean calorie content of the chocolate energy bars, we will use the sample mean, the sample standard deviation, and the t-distribution since the sample size is small. Given are the sample mean (×) is 230 calories, the sample standard deviation (s) is 15 calories, and the sample size (n) is 10.

Steps to follow:

Identify the appropriate t-value for the 99% CI, which corresponds to a two-tailed test with 9 degrees of freedom (n-1). From the t-distribution table, this value is approximately 3.25.

Calculate the standard error (SE) of the mean by dividing the standard deviation by the square root of the sample size: SE = s / √n = 15 / √10 ≈ 4.74.

Multiply the t-value by the SE to get the margin of error (ME): ME = t * SE ≈ 3.25 * 4.74 ≈ 15.41.

Finally, subtract and add the ME from the sample mean to get the lower and upper bounds of the CI: (× - ME, × + ME) = (230 - 15.41, 230 + 15.41) = (214.59, 245.41).

Therefore, the 99% confidence interval for the true mean calorie content is approximately (214.59, 245.41) calories.

Explanation of a 95% CI: A 95% confidence interval means that if we were to take 100 different random samples from the population and construct a CI for each using the same method, approximately 95 of these intervals would contain the true population mean.

Answer:

The probability that their mean length is less than 11.2 inches is 0.5987

Step-by-step explanation:

Mean = 10.9 inches

Standard deviation = 1.2 inches

We are supposed to find If 25 items are chosen at random, what is the probability that their mean length is less than 11.2 inches

Formula : [tex]Z=\frac{x-\mu}{\sigma}[/tex]

We are supposed to find P(x<11.2)

[tex]Z=\frac{11.2-10.9}{1.2}[/tex]

[tex]Z=0.25[/tex]

Refer the z table for p value

p value = 0.5987

Hence the probability that their mean length is less than 11.2 inches is 0.5987

Answer:

the area of the triangle is 4 square units.

Step-by-step explanation:

Plotting the points, we can see that the triangle is isosceles lying in the 4th quadrant of graph.

we can break the triangle in 2 similar right angled triangles,

each with base 2 and height 2  units.

area of triangle is given by the formula,

A= [tex](\frac{1}{2})(base)(height)[/tex]

thus, A= [tex](\frac{1}{2})(2)(2)[/tex]

A=2 square units.

there are 2 such triangles,

thus total area is 4 square units.

Answer:

a) $342,491

$342.491

$389.74

b) $400

c) $320

Step-by-step explanation:

the cost function = C(x)

C(x) = 16000 + 200x + 4x^3/2

a) when we have a unit of 1000 unit, x= 1000

C(1000) = 16000 + 200(1000) + 4(1000)^3/2

= 16000 + 200000 + 126491

= 342,491

Cost = $342,491

Average cost= C(1000) / 1000

= 342,491/1000

= 342.491

The average cost = $342.491

Marginal cost = derivative of the cost

C'(x) = 200 + 4(3/2) x^1/2

= 200 + 6x^1/2

C'(1000) = 200 + 6(1000)^1/2

= 389.74

Marginal cost = $389.74

Marginal cost = Marginal revenue

C'(x) = C(x) / x

200 + 6x^1/2 = (16000 + 200x + 4x^3/2) / x

200 + 6x^1*2 = 16000/x + 200 +4x^1/2

Collect like terms

6x^1*2 - 4x^1/2 = 16000/x + 200 -200

2x^1/2 = 16000/x

2x^3/2 = 16000

x^3/2 = 16000/2

x^3/2 = 8000

x = 8000^2/3

x = 400

Therefore, the production level that will minimize the average cost is the critical value = $400

C'(x) = C(x) / x

C'(400) = 16000/400 + 200 + 4(400)^1/2

= 40 + 200 + 80

= 320

The minimum average cost = $320

Answer: These events are dependent.

Step-by-step explanation: The probability of the second woman getting a pink rose is affected by the first woman getting a pink rose as the pink rose obtained by the first woman was not replaced. Hence there are less pink roses in the flower vase and hence lower probability that the second woman gets a pink rose. These events are thus dependent.

Answer:

Dependent because when the first flower is taken, it affects the ratio of the types of flowers in the vase.

Step-by-step explanation:

Answer:

336.02 square centimeters

Step-by-step explanation:

The surface area is the area of all the surfaces of the prism shown.

The prism has 7 surfaces.

Top and Bottom are pentagons with side lengths of 5.

The other 5 side surfaces are rectangles with length 10 and width 5.

Note the formulas of area of pentagon and area of rectangle below:

Area of Rectangle = Length * Width

Area of Pentagon = [tex]\frac{1}{4}\sqrt{25+10\sqrt{5} }* a^2[/tex] ,  where a is the side length

Lets find area of each of the surfaces:

Top Surface (Pentagon with side length 5) = [tex]\frac{1}{4}\sqrt{25+10\sqrt{5} }* a^2=\frac{1}{4}\sqrt{25+10\sqrt{5} }* (5)^2=43.01[/tex]

Bottom Surface = same as Top Surface = 43.01

Side Surface (rectangle with length 10 and width 5) = 10 * 5 = 50

There are 5 side surfaces that are each 50 sq. cm. so area would be:

Area of 5 Side Surface = 5 * 50 = 250

Total Surface Area = 250 + 43.01 + 43.01 = 336.02 square centimeters

Answer:

The dimensions of the pen that minimize the cost of fencing are:

[tex]x \approx 12.25 \:ft[/tex]

[tex]y \approx 8.17 \:ft[/tex]

Step-by-step explanation:

Let [tex]x[/tex] be the width and [tex]y[/tex] the length of the rectangular pen.

We know that the area of this rectangle is going to be [tex]x\cdot y[/tex].The problem tells us that the area is 100 feet, so we get the constraint equation:

[tex]x\cdot y=100[/tex]

The quantity we want to optimize is going to be the cost to make our fence. If we have chain-link on three sides of the pen, say one side of length [tex]y[/tex] and both sides of length [tex]x[/tex], the cost for these sides will be

[tex]10(y+2x)[/tex]

and the remaining side will be fence and hence have cost

[tex]20y[/tex]

Thus we have the objective equation:

[tex]C=10(y+2x)+20y\\C=10y+20x+20y\\C=30y+20x[/tex]

We can solve the constraint equation for one of the variables to get:

[tex]x\cdot y=100\\y=\frac{100}{x}[/tex]

Thus, we get the cost equation in terms of one variable:

[tex]C=30(\frac{100}{x})+20x\\C=\frac{3000}{x}+20x[/tex]

We want to find the dimensions that minimize the cost of the pen, for this reason, we take the derivative of the cost equation and set it equal to zero.

[tex]\frac{d}{dx} C=\frac{d}{dx} (\frac{3000}{x}+20x)\\\\\mathrm{Apply\:the\:Sum/Difference\:Rule}:\quad \left(f\pm g\right)'=f\:'\pm g'\\\\C'(x)=\frac{d}{dx}\left(\frac{3000}{x}\right)+\frac{d}{dx}\left(20x\right)\\\\C'(x)=-\frac{3000}{x^2}+20[/tex]

[tex]C'(x)=-\frac{3000}{x^2}+20=0\\\\-\frac{3000}{x^2}x^2+20x^2=0\cdot \:x^2\\-3000+20x^2=0\\-3000+20x^2+3000=0+3000\\20x^2=3000\\\frac{20x^2}{20}=\frac{3000}{20}\\x^2=150\\\\\mathrm{For\:}x^2=f\left(a\right)\mathrm{\:the\:solutions\:are\:}x=\sqrt{f\left(a\right)},\:\:-\sqrt{f\left(a\right)}\\\\x=\sqrt{150},\:x=-\sqrt{150}[/tex]

Because length must always be zero or positive we take [tex]x=\sqrt{150}[/tex] as only value for the width.

To check that this is indeed a value of [tex]x[/tex] that gives us a minimum, we need to take the second derivative of our cost function.

[tex]\frac{d}{dx} C'(x)=\frac{d}{dx} (-\frac{3000}{x^2}+20)\\\\C''(x)=-\frac{d}{dx}\left(\frac{3000}{x^2}\right)+\frac{d}{dx}\left(20\right)\\\\C''(x)=\frac{6000}{x^3}[/tex]

Because [tex]C''(\sqrt{150})=\frac{6000}{\left(\sqrt{150}\right)^3}=\frac{4\sqrt{6}}{3}[/tex] is greater than zero, [tex]x=\sqrt{150}[/tex] is a minimum.

Now, we need values of both x and y, thus as [tex]y=\frac{100}{x}[/tex], we get

[tex]x=\sqrt{150}=5\sqrt{6}=12.25[/tex]

[tex]y=\frac{100}{\sqrt{150}}=\frac{10\sqrt{6}}{3}\approx 8.17[/tex]

The dimensions of the pen that minimize the cost of fencing are:

[tex]x \approx 12.25 \:ft[/tex]

[tex]y \approx 8.17 \:ft[/tex]

Answer:

  yes

Step-by-step explanation:

The graph passes the vertical line test, so is the graph of a function (yes). Each input value has exactly one output value.