Math  /  Algebra

QuestionA fence is to be built to enclose a rectangular area of 250 square feet. The fence along three sides is to be made of material that costs : dollars per foot, and the material for the fourth side costs 14 dollars per foot. Find the dimensions of the enclosure that is most economical to construct.
Dimensions: \square x \square
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Studdy Solution

STEP 1

1. The rectangular area to be enclosed is 250 square feet.
2. Three sides of the fence use material costing x x dollars per foot.
3. The fourth side uses material costing 14 dollars per foot.
4. We aim to minimize the total cost of the fence.

STEP 2

1. Define variables and expressions for the dimensions and cost.
2. Set up the equation for the area.
3. Express the cost function in terms of one variable.
4. Use calculus to find the minimum cost.
5. Solve for the dimensions.

STEP 3

Let the length of the side with the different cost be L L and the width be W W .
The cost for the three sides is 2Wx+Lx 2W \cdot x + L \cdot x .
The cost for the fourth side is L14 L \cdot 14 .

STEP 4

The area equation is given by:
LW=250 L \cdot W = 250

STEP 5

Express L L in terms of W W using the area equation:
L=250W L = \frac{250}{W}
Substitute L L into the cost function:
Total Cost C=2Wx+250Wx+250W14 C = 2W \cdot x + \frac{250}{W} \cdot x + \frac{250}{W} \cdot 14
Simplify the cost function:
C=2Wx+250xW+3500W C = 2Wx + \frac{250x}{W} + \frac{3500}{W}

STEP 6

To find the minimum cost, take the derivative of C C with respect to W W and set it to zero:
dCdW=2x250xW23500W2=0 \frac{dC}{dW} = 2x - \frac{250x}{W^2} - \frac{3500}{W^2} = 0
Simplify and solve for W W :
2x=250x+3500W2 2x = \frac{250x + 3500}{W^2}
2xW2=250x+3500 2xW^2 = 250x + 3500
W2=250x+35002x W^2 = \frac{250x + 3500}{2x}
W=250x+35002x W = \sqrt{\frac{250x + 3500}{2x}}

STEP 7

Substitute W W back into the expression for L L :
L=250W L = \frac{250}{W}
Calculate the dimensions using the expression for W W .
Dimensions: W=250x+35002x W = \sqrt{\frac{250x + 3500}{2x}} , L=250250x+35002x L = \frac{250}{\sqrt{\frac{250x + 3500}{2x}}}

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