Answer to Question #198922 in Calculus for rameesha

Question

A small rectangular warehouse is to be constructed which is to have an area of 10000 square feet. The building is to be partitioning internally in to eight equal parts. The costs have been estimated based on exterior and interior walls dimensions. The costs are $200 per running foot of exterior wall and plus $100 per running foot of interior wall. [06] a) Determine the dimensions which will minimize the construction costs? b) What are the minimum costs?

Answer 1

(a)

The figure is shown as:

Let,

Length of the rectangle = x ft

Width of the rectangle = y ft

we can see that, there are 7 interior walls of length y.

Total length of interior walls = 7y ft

Total length of the exterior walls = x+x+y+y = (2x+2y) ft

Construction cost of exterior walls = 200(2x+2y) dollars

Construction cost of interior walls = 100(7y) dollars

we know that,

∴Area=xy

Area of rectangle=length×width

∴Area=xy

Given,

A=10000

⇒xy=10000

$⇒y=\frac{10000} {x} ⋯ (1)$

Now,

Total Construction cost = construction cost of exterior walls + construction cost of interior walls

C=200(2x+2y)+100(7y)

C=400x+400y+700y

C=400x+1100y

Substituting value of y from equation 1,

$C=400x+1100(\frac{10000} {x} )$

$C=400x+\frac{11000000}{ x}$ ⋯ (2)

$C'=400(1)[phone](−x ^{-2}−2 )$

To minimize the construction costs, differentiate the cost function with respect to x and put it equal to zero and find critical point.

$C'=\frac{d}{dx}(400x+\frac{11000000}{x})$

$C'=400\frac{d}{ dx} (x)[phone]\frac{d}{ dx} (x ^{−1} )$

$C'=400(1)[phone](−x ^{−2} )$

$C'=400−\frac{11000000}{x^2}$

$400x^2=1100000$

$x^2=27500$

$x=50\sqrt{11}$

$x\approx165.83ft$

From equation 1

$y=\frac{10000}{x}$

$y=\frac{10000}{165.83}$

$y\approx60.83 ft$

(b)

The minimum costs are:

$Cost \space of \space exterior \space walls\\=200(2x+2y)\\=400(x+y)\\=400(165.83+60.3)\\=400(226.13)\\=90452 \space dollars$

Cost of interior walls

$=100(7y)\\=700y\\=700(60.3)\\=42210 \space dollars$

Total minimum cost$=90452+42210\\=132662\space dollars$