   Chapter 4.7, Problem 23E Calculus: Early Transcendental Fun...

7th Edition
Ron Larson + 1 other
ISBN: 9781337552516

Solutions

Chapter
Section Calculus: Early Transcendental Fun...

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

Minimum Length and Minimum Area A right triangle is formed in the first quadrant by the x- and y-axes and a line through the point (1, 2) (see figure).(a) Write the length L of the hypotenuse as a function of x.(b) Use a graphing utility to approximate x graphically such that the length of the hypotenuse is a minimum.(c) Find the vertices of the triangle such that its area is a minimum. (a)

To determine

To calculate: The hypotenuse’s length as a function of the variable x.

Explanation

Given:

A right angled triangle is formed in the first quadrant between the x-axis, the y-axis and the line passing through the point (1,2).

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:

Let the line intersect the y-axis at (0,y) and the x-axis at the point (x,0).

First obtain the equation of the line passing through the points (1,2) and (x,0)

(b)

To determine

To graph: The length function obtained to see the point at which it is minimum.

(c)

To determine

To calculate: The vertices of the triangle such that its area is minimum.

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