1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
Olin [163]
2 years ago
14

A farmer wants to fence in a rectangular plot of land adjacent to the north wall of his barn. No fencing is needed along the bar

n, and the fencing along the west side of the plot is shared with a neighbor who will split the cost of that portion of the fence. If the fencing costs $8 per linear foot to install and the farmer is not willing to spend more than $4000, find the dimensions for the plot that would enclose the most area.
Mathematics
1 answer:
Luba_88 [7]2 years ago
8 0

The dimensions for the plot that would enclose the most area are a length and a width of 125 feet.

In this question we shall use the first and second derivative tests to determine the <em>optimal</em> dimensions of a rectangular plot of land. The perimeter (p), in feet, and the area of the rectangular plot (A), in square feet, of land are described below:

p = 2\cdot (w+l) (1)

A = w\cdot l (2)

Where:

  • w - Width, in feet.
  • l - Length, in feet.

In addition, the cost of fencing of the rectangular plot (C), in monetary units, is:

C = c\cdot p (3)

Where c is the fencing unit cost, in monetary units per foot.

Now we apply (2) and (3) in (1):

p = 2\cdot \left(\frac{A}{l}+l \right)

\frac{C}{c} = 2\cdot (\frac{A}{l}+l )

\frac{C\cdot l}{c} = 2\cdot (A+l^{2})

\frac{C\cdot l}{c}-2\cdot l^{2} = 2\cdot A

\frac{C\cdot l}{2\cdot c} - l^{2} = A (4)

We notice that fencing costs are directly proportional to the area to be fenced. Let suppose that cost is the <em>maximum allowable </em>and we proceed to perform the first and second derivative tests:

FDT

\frac{C}{2\cdot c}-2\cdot l = 0

l = \frac{C}{4\cdot c}

SDT

A'' = -2

Which means that length leads to a <em>maximum</em> area.

If we know that c = 8 and C = 4000, then the dimensions of the rectangular plot of land are, respectively:

l = \frac{4000}{4\cdot (8)}

l = 125\,ft

A = \frac{(4000)\cdot (125)}{2\cdot (8)} -125^{2}

A = 15625\,ft^{2}

w = \frac{15625\,ft^{2}}{125\,ft}

w = 125\,ft

The dimensions for the plot that would enclose the most area are a length and a width of 125 feet.

We kindly invite to check this question on areas: brainly.com/question/11952845

You might be interested in
What is the solution to the quadratic equation?<br> y = x^2-4
Anika [276]

(-2,2)

i just did my homework with the same question

7 0
2 years ago
Find the measure of each interior angle and each exterior angle of the indicated regular polygon. 25-gon
Katena32 [7]

Each exterior angle of a regular 25-gon = 360/25 = 14.4 deg. and each interior angle= 165.6 deg.

6 0
2 years ago
Read 2 more answers
Jackie found the area of a square to be 25/16 square feet.Which shows the side length of the square?
svlad2 [7]

The side length is the square root of the area, so is ...

... √(25/16 ft²) = (√25)/(√16) ft = 5/4 ft

4 0
3 years ago
Read 2 more answers
Write the equation of the line that goes through (-2, 0) and (4,3).
Snezhnost [94]

Answer:

B

Step-by-step explanation:

I just put in in the desmos graphing calculator

8 0
2 years ago
There are 153 stickers and 37 bouncy balls in the prize bin. If each child takes 10
Darya [45]

Answer:

19 prizes.

Step-by-step explanation:

First add 153 and 37 which is 190. Then divide 190 and 10. Which is 19.

7 0
3 years ago
Other questions:
  • "Why do whole numbers raised to an exponent get greater, while fractions raised to an exponent get smaller?" I completely unders
    9·1 answer
  • Evaluate the expression when a=2 and x= -3
    13·1 answer
  • A population of 200 animals has a growth rate of 1.03% each year. at this growth rate, the function f(x)=200(1.03)^x gives the p
    6·1 answer
  • On a blueprint, a square with 20 meters per side was shown as a square with 12 centimeters per side. how long would a girder app
    6·1 answer
  • What changes in the parent graph y= |x| would you see if you graphed y= |x+3| ?
    5·2 answers
  • Solution for 11m-8m=
    6·2 answers
  • You meet with the financial aid office to discuss your costs for attending MGA next semester. Tuition is
    5·1 answer
  • Identify the property shown by the equation.<br> 14 × 6 = 6 × 14
    10·1 answer
  • Find all of the missing angle measures. Remember you cannot assume right angles or diameters. Also think about how many degrees
    5·1 answer
  • Write the standard number.<br><br>3.6 * 10^-6​
    11·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!