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belka [17]
3 years ago
8

A rancher wants to create two rectangular pens, as shown in the figure, using an existing fence line as one side. The pens need

to have a total area of 972 square feet. What dimensions should be used to minimize the amount of fence used?

Mathematics
1 answer:
ololo11 [35]3 years ago
6 0
<span>A rancher wants to create two rectangular pens, as shown in the attached figure

Let the side parallel to the </span><span>existing fence line = y
And the sides which is perpendicular to </span><span>existing fence line = x
</span><span>The pens need to have a total area of 972 square feet
</span>∴ x y = 972 
∴ y = 972/x

Let the length of needed fence = L
∴ L = y + 3x
substituting with the value of y
∴ L = \frac{972}{x} +3x
differentiating the length with respect to x and equating with zero
∴ \frac{dL}{dx} =  \frac{-972}{x^2} +3 = 0
solve for x
∴ x = 18
substituting to find y
∴ y = 54

<span>The dimensions should be used to minimize the amount of fence used
is 18 , 54 feet </span>

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