BuyFind

Precalculus: Mathematics for Calcu...

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

Precalculus: Mathematics for Calcu...

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

Solutions

Chapter 1.5, Problem 118E
To determine

The distance from the center of earth to the dead spot.

Expert Solution

Answer to Problem 118E

The distance from the center of earth to the dead spot is 215000mi_.

Explanation of Solution

Given:

The net gravitational force F acting on an object situated on imaginary line segment is drawn between the centers of the earth and the moon is F=Kx2+0.012K(239x)2, where K>0 is a constant and x is the distance of the object from the center of the earth, measured in thousands of miles.

Formula used:

Quadratic formula:

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:

Note that, no net gravitational force act between centre of the earth and dead spot. Therefore, F=0.

Substitute 0 for F in the equation F=Kx2+0.012K(239x)2.

0=Kx2+0.012K(239x)2Kx2=0.012K(239x)2K(239x)2=0.012Kx2K(239x)20.012Kx2=0

Simplify the above equation as follows.

K(239x)20.012Kx2=0(239x)20.012x2=057121478x+x20.012x2=00.988x2478x+57121=0

Use the Quadratic formula to find the value of x.

Substitute 0.988 for a, 478 for b and 57121 for c in the quadratic formula.

x=(478)±(478)24×0.988×(57121)2×0.988=478±228484225742.1921.976=478±2741.8081.976=478±52.3621.976

Simplify further as follows.

x=478+52.3621.976or x=47852.3621.976x=530.3621.976or x=425.6381.976x268or x215

Note that, 268(000) miles is greater than the distance from the earth to the moon.

Thus, the distance from the center of earth to the dead spot is 215000mi_.

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