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 112E
To determine

The speed of the ball at which it should be thrown upward to reach a maximum height of 100ft.

Expert Solution

Answer to Problem 112E

The speed of the ball at which it should be thrown upward to reach a maximum height of 100ft is 80ft/s_.

Explanation of Solution

Given:

An object thrown or fired straight upward at an initial speed of v0 ft/s will reach a height of h feet after t seconds, where h and t are related by the formula h=16t2+v0t.

Result used:

Discriminant of a quadratic equation:

The quantity b24ac is called the discriminant of a quadratic equation of the form ax2+bx+c=0,a0, from which the number of solutions the quadratic equation can have, can be obtained.

1. If b24ac>0, there are two unequal real solutions.

2. If b24ac=0, there is a repeated real solution, a double root.

3. If b24ac<0, there is no real solution.

Calculation:

Substitute h=100 in h=16t2+v0t and simplify the equation as 16t2v0t+100=0.

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

Here a=16,b=v0and c=100.

Note that, the ball will reach the highest point only once and the quadratic equation has only one solution when the discriminant is zero.

Therefore, let the discriminant be zero.

D=b24ac0=(v0)24×16×1000=v026400v02=6400

On further simplifications, the following is obtained.

v0=6400v0=±80

Since the speed should be always positive, the value of v0=80.

Thus, the speed of the ball at which it should be thrown upward to reach a maximum height of 100ft is 80ft/s_.

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