Concept explainers
Falling-Body Problems Use the formula h = − 16t2 + v0t discussed in Example 9.
129. A ball is thrown straight upward at an initial speed of v0 = 40 ft/s.
- (a) When does the ball reach a height of 24 ft?
- (b) When does it reach a height of 48 ft?
- (c) What is the greatest height reached by the ball?
- (d) When does the ball reach the highest point of its path?
- (e) When does the ball hit the ground?
(a)
The time taken by the ball to reach a height of
Answer to Problem 111E
The ball reaches a height of
Explanation of Solution
Given:
An object thrown or fired straight upward at an initial speed of
Calculation:
The height of the ball is given as
Therefore, substitute
Simplify further as follows.
On further simplifications, the following is obtained.
Thus, the ball reaches a height of
(b)
The time taken by the ball to reach a height of
Answer to Problem 111E
The ball
Explanation of Solution
Formula used:
Quadratic formula:
The solution of a quadratic equation of the form
Calculation:
The height of the ball is given as
Therefore, substitute
Simplify further as follows.
Compare this equation with the general form
Here
Use the quadratic formula to obtain the roots of the given equation.
Since the square of any real number is nonnegative,
Thus, the ball
(c)
The greatest height reached by the ball.
Answer to Problem 111E
The greatest height reached by the ball is
Explanation of Solution
Result used:
Discriminant of a quadratic equation:
The quantity
1. If
2. If
3. If
Calculation:
Given that the height of an object thrown straight upward at an initial speed of
Since the ball is thrown straight upward at an initial speed of
Compare this equation with the general form
Here
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.
Simplify the above equation as follows.
Thus, the greatest height reached by the ball is
(d)
The time taken by the ball to reach the highest point.
Answer to Problem 111E
The ball reaches the highest point after
Explanation of Solution
The greatest height reached by the ball is obtained as
Substitute
Compare this equation with the general form
Here
Use the quadratic formula to obtain the roots of the given equation.
Simplify the above equation as follows.
Thus, the ball reaches the highest point after
(e)
The time taken by the ball to hit the ground.
Answer to Problem 111E
The ball hits the ground after
Explanation of Solution
The height of a ball is
Substitute
Simplify further as follows.
That is,
Thus, the ball hits the ground after
Chapter 1 Solutions
Precalculus: Mathematics for Calculus - 6th Edition
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