BuyFind

Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805
BuyFind

Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805

Solutions

Chapter 3.5, Problem 56E
To determine

To find: How far above x-axis is the lamp located.

Expert Solution

Answer to Problem 56E

The lamp is located 2 units above the x-axis.

Explanation of Solution

Given:

The lamp located three unit from the right of y axis,

The shadow created by elliptical region x2+4y25.

The point (5,0) is on the edge of the shadow.

Calculation:

Let h be the height of the lamp.

Let (a,b) be an point of the tangency through the point (3,h) and (5,0).

The slope of the line passing through the point (3,h) and (5,0),

m=0h53=h8

Thus, the slope of the tangent at (a,b) is m=h8 (1)

Consider the equation of the ellipse x2+4y2=5,

Differentiate with respect to x,

ddx(x2)+4ddx(y2)=ddx(5)2x+4(2ydydx)=0dydx=2x8ydydx=x4y

That is, the slope of the tangent to the ellipse is, dydx=x4y.

The slope of the tangent to the ellipse at (a,b) is dydx=a4b (2)

From equations (1) and (2),

h8=a4bh=2ab

The slope of the line passing through the point (a,b) and (5,0),

m=b0a+5

m=ba+5 (3)

Form equations (2) and (3),

a4b=ba+5a25a=4b2a2+4b2=5a

Substitute (a,b) in x2+4y2=5,

a2+4b2=5

Since a2+4b2=5a,

5a=5a=1

Substitute a=1 in a2+4b2=5,

1+4b2=54b2=51b2=1b=1

Therefore, the point (a,b)=(1,1).

Substitute (a,b)=(1,1) in h=4ab,

h=2(1)1=2

Therefore, the lamp is located 2 units above the x-axis.

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