Question

A rectangular block has a square base of side x cm, a height of y cm and a

volume of 80cm3

. The base and top are to be covered with lacquer costing 5

cents/cm2 and the sides with lacquer costing 4 cents /cm2

. Find the values of

x and y for minimum cost.​

Answer 1

Given:

A rectangular block has a square base of side x cm, a height of y cm and a

volume of $80cm^{3}$.

The base and top are to be covered with lacquer costing $5 cents/cm^2$ and the sides with lacquer costing $4 cents /cm2$.

To Find:

Find the values of  x and y for minimum cost ?

Solution:

it is given that - volume of rectangular block is 80 cm^3 having square base of side x cm and height of y cm.

therefore,

$Volume = (area of square base)(height)\\ 80 cm^{3} = (x^{2} )(y)\\x^{2} y=80$

$y=\dfrac{80}{x^{2} }$              .....(1)

Also, given that -  The base and top are to be covered with lacquer costing $5 cents/cm^2$ and the sides with lacquer costing $4 cents /cm2$.

therefore,

Also, area of base= area of top

so, total cost of lacquer to cover base and top with lacquer is ;

$=(x^{2} +x^{2} )(5 ) \\=10x^{2}$.....(2)

so, total cost of lacquer to cover sides is;

$=4(xy)(4)\\=16xy$......(3)

on putting value of y from equation (1) , we get

total cost of lacquer to cover sides is

$=16x(80/x^{2} )\\=1280/x$

therefore, total cost of lacquer to cover rectangular block is

$=10x^{2} +1280/x\\\\=\dfrac{10x^{3}+1280}{x}$