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Of all rectangles with area 324324​, which one has the minimum​ perimeter? Let P and w be the perimeter and​ width, respectively, of the rectangle. Write the objective function in terms of P and w. Assume that the width is less than the length if the dimensions are unequal.

Respuesta :

Answer:

For w = 18 units perimeter is minimum

P = 2(18 + w)

Step-by-step explanation:

Given;

Area of the rectangle = 324 units²

P is the perimeter

w is the width

Let L be the length of the rectangle

therefore,

P = 2(L + w)  ............(1)

also,

Lw = 324

or

L = [tex]\frac{324}{w}[/tex] ..........(2)

substituting 2 in 1

P = [tex]2(\frac{324}{w} + w)[/tex]

now,

for minimizing the perimeter

[tex]\frac{dp}{dw}=\frac{d(2(\frac{324}{w} + w))}{dw}[/tex] = 0

or

[tex]2((-1)\frac{324}{w^2}+1)[/tex] = 0

or

[tex](-1)\frac{324}{w^2}+1[/tex] = 0

or

[tex](-1)\frac{324}{w^2}[/tex] = -1

or

w² = 324

or

w = 18 units

For w = 18 units perimeter is minimum

therefore,

from 2

L = [tex]\frac{324}{18}[/tex]

or

L = 18 units

objective function for P is:

P = 2(18 + w)

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