# The distance of the object from the lens. ### Precalculus: Mathematics for Calcu...

6th Edition
Stewart + 5 others
Publisher: Cengage Learning
ISBN: 9780840068071 ### Precalculus: Mathematics for Calcu...

6th Edition
Stewart + 5 others
Publisher: Cengage Learning
ISBN: 9780840068071

#### Solutions

Chapter 1.5, Problem 114E
To determine

## The distance of the object from the lens.

Expert Solution

The distance of the object from the lens is 12cm_.

### Explanation of Solution

Given:

The equation that relates F, x and y is 1F=1x+1y, where F is the focal length of the convex lens, x is the distance of the object from the lens and y is the  distance of the image from the lens.

Formula used:

The solution of a quadratic equation of the form ax2+bx+c=0,a0 can be obtained by using the quadratic formula x=b±b24ac2a.

Calculation:

The focal length of convex lens is 4.8cm.

Since the image of an object is 4 cm closer to the lens than the object itself, it can be written that y=x4.

Substitute 4.8 for F and x4 for y in the equation 1F=1x+1y.

14.8=1x+1x414.8=x4+xx(x4)14.8=2x4x(x4)x(x4)=4.8(2x4)

Simplify the above equation as follows.

x24x=9.6x19.2x24x9.6x+19.2=0x213.6x+19.2=0

Compare this equation with the general form ax2+bx+c=0.

Here a=1,b=13.6and c=19.

Use the quadratic formula to obtain the roots of the given equation.

x=(13.6)±(13.6)24×1×192×1=13.6±184.96762=13.6±108.962=13.6±10.42

Simplify further as follows.

x=13.6+10.42or x=13.610.42x=242or x=3.22x=12or x=1.6

Substitute x=12 in y=x4.

y=124=8

Substitute x=1.6 in y=x4.

y=1.64=2.4

Note that, the distance cannot be negative.

Thus, the distance of the object from the lens is 12cm_.

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