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31.12.2019 •
Mathematics
Afence is to be built to enclose a rectangular area of 320 square feet. the fence along three sides is to be made of material that costs 5 dollars per foot, and the material for the fourth side costs 12 dollars per foot. find the dimensions of the enclosure that is most economical to construct.
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Ответ:
We know:
A = 320ft²
3 fences - 5 dollars per foot and 1 fence - 12 dollars per foot.
A = 320 = xy
Eliminate one variable and solve for that:
x =
We need to now find an equation for the amount of money we are going to spend.
We can visualise it in our heads: We have two short sides to be built and two long sides to be built. One side will cost us 12 dollars, while the other three costs just 5 dollars to build.
We can now find our perimeter, which is given by 2x + 2y, but we know that x can be replaced as
Hence, our perimeter becomes: 2(
Now, let's write an equation for C(y), where C is our total cost.
C(y) = 5*(2*
Now, we want to find when it's a minimum so that it will cost us the least amount of money.
Take the first derivative, and let it equal to 0.
C'(y) = -3200*y⁻² + 17 = 0
3200 = 17y²
y² =
y =
Now, to prove that it's a min. at y =
C''(y) = 3200y⁻³ and at y =
So, at y =
Substitute y, back into (1) to find for x.
x =
therefore, x =
Ответ: