Question

The top and bottom margins of a poster are each 9 cm and the side margins are each 6 cm. If the area of the printed material on the poster is fixed at 864 cm2, find the dimensions of the poster with the smallest area.

Answer 1

Answer:

the dimensions of the poster with the smallest area is 36cm by 42cm

Step by step Explanation:

Let us denote the width and height of the printed area as x and y respectively.

Then xy=864cm^2

y= 864/x

The total height of the poster , along with 9cm margin at sides is (y+18)

The total width of the poster , along with 6cm margin at sides is (y+12)

the area of the total poster is

A= (x+12)(y+18)...............eqn(1)

Substitute the value of y into equation (1)

A= (x+12)[(⁸⁶⁴/ₓ)+(18)]

864+ 18x +¹⁰³⁶⁸/x +216

18x +( ⁰³⁶⁸/x) + 1080

then find the derivative of x in this equation by

A'= 18-(¹⁰³⁶⁸/x²)

then equate A' to be zero so as to find the value of x

0= 18- (¹⁰³⁶⁸/x²)

18= (¹⁰³⁶⁸/x²)

x²= 576

x= √576

x= 24

Since A'' is positive for x> 0 then A is concave up and x= 24 I s the minimum value, the value of y as well is

y= 864/x

y= 432

Then the width of the poster is now

x+12= 24+ 12= 36

The corresponding height is

y+8= 24+18= 42

Therefore,the dimensions of the poster with the smallest area is 36cm by 42cm