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
Find the LCM of 98 and 182.
LUCKY_DIMON [66]

Answer:

LCM = 1274

Step-by-step explanation:

98 = 2 × 7²

182 = 2 × 7 × 13

LCM(98 ; 182) = 2 × 72 × 13 = 1274

5 0
3 years ago
Write a real world situation of |x-14|=3
natita [175]
X -14 +14={3+14}
×=17
answer is 17 .
7 0
3 years ago
If sample proportion is 0.476 and margin of error is given as 0.044 then find the respective confidence interval
torisob [31]

Answer:

Step-by-step explanation:

4 0
3 years ago
Help Pls if you want!!!!
Tom [10]

Answer:

Its 2.5 percent ladd enjoy

Step-by-step explanation:

7 0
3 years ago
Read 2 more answers
How do you turn 72.1 repeating number into a fraction?
SCORPION-xisa [38]
72 1/9

0.1 recurring of the number 1 is 1/9 of the number 
5 0
3 years ago
Other questions:
  • A 100 ft. ladder rests on top of a hook and ladder truck with its base 11 feet from the ground. When the angle of elevation of t
    10·1 answer
  • solve this system of linear equations. separate the x- and y-values with a comma -2x = 10 -6y, 20x = 62 +6y
    6·1 answer
  • 100 POINTS please answer! ill give you brainliest but i rlly need help! (the circled ones!)
    7·2 answers
  • -20 , -16 , -12 , -8 ....
    11·1 answer
  • Diego and Max are buying soft drinks for a neighborhood picnic. Each person is
    6·1 answer
  • If possible can someone help? Thank you .
    7·1 answer
  • Carlos mixed 3 1⁄2 cups of red paint with 4 1⁄2 cups of yellow paint to make an orange paint. How many cups of red paint and how
    8·1 answer
  • If a = -2 and b = -4 -3i, then find the value of a b in fully simplified form.​
    12·2 answers
  • Find the value of Y<br><br> Help plz whenever you can :D
    12·1 answer
  • F) m(4+p) +p(3 + m)
    5·2 answers
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!