Question

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.

Answer 1

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 = $\frac{324}{w}$ ..........(2)

substituting 2 in 1

P = $2(\frac{324}{w} + w)$

now,

for minimizing the perimeter

$\frac{dp}{dw}=\frac{d(2(\frac{324}{w} + w))}{dw}$ = 0

or

$2((-1)\frac{324}{w^2}+1)$ = 0

or

$(-1)\frac{324}{w^2}+1$ = 0

or

$(-1)\frac{324}{w^2}$ = -1

or

w² = 324

or

w = 18 units

For w = 18 units perimeter is minimum

therefore,

from 2

L = $\frac{324}{18}$

or

L = 18 units

objective function for P is:

P = 2(18 + w)