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Calculus: Early Transcendental Fun...

7th Edition
Ron Larson + 1 other
ISBN: 9781337552516

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BuyFindarrow_forward

Calculus: Early Transcendental Fun...

7th Edition
Ron Larson + 1 other
ISBN: 9781337552516
Textbook Problem

Maximum Area The figures show a rectangle, a circle, and a semicircle inscribed in a triangle bounded by the coordinate axes and the first-quadrant portion of the line with intercepts (3, 0) and (0, 4). Find the dimensions of each inscribed figure such that its area is maximum. State whether calculus was helpful in finding the required dimensions. Explain your reasoning.

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To determine

To calculate: The dimensions of each inscribed figure that gives the maximum area where:

images

Explanation

Given:

The figure:

Formula used:

For a function f that is twice differentiable on an open interval I, if f'(c)=0 for some c, then,

If f''(c)>0 the function f has relative minima at c if f''(c)<0 the function f has relative maxima at c.

Calculation:

The intercept form of the line in all the three figures is:

x3+y4=1

This can be rewritten as:

y=443x

Now consider the first figure as the rectangle.

The area of the rectangle would be:

A=xy=x(443x)=4x43x2

Now differentiate this with respect to x and equate that to zero to get the critical points.

A'(x)=483x483x=083x=4x=32

Now compute the second derivative at the critical point to get,

A''(x)=8x<0

This implies that the function has a relative maximum at this point.

This gives y as:

y=443(32)=42=2

Hence, the dimensions of the rectangle are 32×2 and calculus was helpful to obtain them

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