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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

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Sphinxa [80]

Answer:

Perimeter: 174.8 m

Area: 1,394 sq m

Step-by-step explanation:

First, the perimeter.

Before we start, let's calculate the circumference of the half-circles at the ends of the field.

The measurement says 2,000 cm, so let's convert it to 20 m for ease.

Circumference of a circle: πd, where d = diameter.. in our case d = 20 m

Circumference of a 20m diameter circle: 20π = 62.8 m

We have 2 half circles... so the perimeter of each half-circle will be: 31.4 m

We also have 800 cm measurement for the "height" of the seating areas... let's convert that in 8 m

We also need to find out the space between the seating area...  We know the whole rectangular pitch is 40 m, then we have to subtract the width of both seating areas (20 and 15 m)... so the space between them is 5m

So, starting with the upper left corner of the rectangular pitch, and working our way clockwise, we encounter the following lengths:

P = 40 + 31.4 + 8 + 10 + 8 + 5 + 8 + 25 + 8 + 31.4 = 174.8 m

The total perimeter is then of 174.8 m

For the area, we need to calculate the area of all forms:

Large rectangular pitch:

LR = 40 x 20 = 800 sq m

The two half circles, form a circle, so A = πr², where r is the radius, which is half the diameter.

AC = π (10)² = 100 π = 314 sq m

Then the seating areas:

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SA2 = 10 x 8 - 80 sq m

Then, we add up everything:

TA = LR + AC + SA1 + SA2

TA = 800 + 314 + 200 + 80 = 1,394 sq m

3 0
3 years ago
If l = 60 inches, w = 65 inches, h = 50 inches, and t = 40 inches, counting the floor, what is the volume of the tent?
defon

Answer:

To find volume of a tent this would be your formula:

start by taking the base of the triangle on your tent typically located in the front

1/2 (height) (width) to get your area (DONT SKIP THIS STEP)

Then take your area you just calculated and do

(area) (length)

Let me give you an example:

in the picture below

height: 3 cm

length: 8 cm

Width: 12 cm

Next do your first formula:

1/2 (3)(12)= 18 cm

(Calculation of one face)

then your second:

(18)(8)= 144 cm^3}

Hopefully this guide helped you sorry i couldn't answer your question without a picture!

5 0
3 years ago
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AysviL [449]

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Step-by-step explanation:

because that is the highest decimal number because it has higher values, I mean "345" is greater than "33", "35", and "332".

the only way the smallest number would be the highest number is when it have a negative symbol to it e.g -0.1 is greater than -0.2 due to the fact that -0.1 is closer to 0,1,2,3,..... than -0.2 According to the number line.

All I'm trying to say is that, if it's not a negative number then the one with the highest value or number is actually the highest number EXCEPT if it Hans the negative symbol (-) to it, then it's the opposite meaning the "smallest number" would be the highest number.

Hope this helps buddy, Good luck ✅.

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