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Calculus (MindTap Course List)

8th Edition
James Stewart
ISBN: 9781285740621

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Calculus (MindTap Course List)

8th Edition
James Stewart
ISBN: 9781285740621
Textbook Problem

A poster is to have an area of 180 in 2 with 1-inch margins at the bottom and sides and a 2-inch margin at the top. What dimensions will give the largest printed area?

To determine

To find:

The dimensions of poster when the area of the poster is maximum

Explanation

1) Concept:

First derivative test:

Suppose c is a critical number of a continuous function f defined on an interval.

(a) If f'(x)>0 for all x<c and f'(x)<0 for all x>c, then f(c) is the absolute maximum value of f.

(b) If f'(x)<0 for all x<c and f'(x)>0 for all x>c, then f(c) is the absolute minimum value of f.

2) Formula:

Area of rectangle l·w

Where l length and w width of rectangle

3) Calculation:

To find the dimensions of poster when area of poster is maximum, assume x and y are width and length of the poster when printed area of poster maximizes

Area of poster is 180 in2

The poster is rectangular in shape.

Area of printed part of poster is y·x

180=y·x

Divide by y

x=180y(1)

The poster has 2 inch margin at top and 1 inch margin at bottom and the sides.

From the graph, the length of printed part of poster is y-2

And width of the printed part of poster is x-3

The poster is rectangle in shape.

Area of printed part of poster is A=y-2(x-3)

From (1) (substitute the value for x),

Ay=y-2180y-3

By multiplication,

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