   Chapter 3.7, Problem 54E

Chapter
Section
Textbook Problem

# Find an equation of the line through the point (3, 5) that cuts off the least area from the first quadrant.

To determine

To find:

An equation of the line through the point (3, 5) that cuts off the least area from the first quadrant

Explanation

1) Concept:

Equation of the line through the point (x1, y1) is given by

y-y1=m(x-x1)

2) Formula:

Area cut off in the first quadrant =12·b·h

3) Calculation:

Equation of the line through the point (3, 5) is

y-5=m(x-3)

Simplify bracket

y-5=mx-3m

To write equation of line in Intercept form

y-5+5=mx-3m+5

Simplify

y= mx-3m+5

Here, x- intercept is 3m-5m and y- intercept is -3m+5

Area cut off in the first quadrant =12·b·h

A(m)=12·(x-intercept )·(y-intercept)

=12·(3m-5m )·(-3m+5)

Simplify

=-3m-522m

Simplify bracket term

=-9m2-30m+252m

Simplify

=-9m2+15-252m

Differentiate

A'm=-912+0+252m2

Simplify

=-92+252m2

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