A ball is dropped from the top of a building that is 1,000 feet high. Its height, in feet, as a function of the time, [tex]x[/tex], in seconds, after the ball was dropped, is given by the following equation:

[tex]f(x) = 1,000 - 16x^2[/tex]

Which set of numbers is appropriate as the domain for this function?

A. Positive Integers
B. Positive Real Numbers
C. Natural Numbers
D. Positive Rational Numbers



Answer :

To determine the appropriate domain for the function [tex]\( f(x) = 1,000 - 16x^2 \)[/tex], let’s go through a step-by-step process:

1. Understand the Function:
- The function represents the height of a ball that is dropped from 1,000 feet.
- Here, [tex]\( x \)[/tex] represents time in seconds.

2. Identify When the Ball Hits the Ground:
- The ball hits the ground when the height [tex]\( f(x) \)[/tex] equals 0.
- Set [tex]\( f(x) = 0 \)[/tex]:
[tex]\[ 1,000 - 16x^2 = 0 \][/tex]
- Solve for [tex]\( x \)[/tex]:
[tex]\[ 16x^2 = 1,000 \\ x^2 = \frac{1,000}{16} \\ x^2 = 62.5 \\ x = \sqrt{62.5} \approx 7.91 \][/tex]

3. Determine the Domain:
- Time [tex]\( x \)[/tex] starts from 0 (when the ball is dropped) and goes up to the point where the ball hits the ground, which is approximately [tex]\( 7.91 \)[/tex] seconds.
- Thus, [tex]\( x \)[/tex] ranges from 0 to approximately [tex]\( 7.905694150420948 \)[/tex].

4. Select the Appropriate Set of Numbers:
- The domain should include all possible values of [tex]\( x \)[/tex] from 0 to approximately 7.91 seconds.
- The most inclusive set that fits this criteria is the Positive Real Numbers, as it includes all values between [tex]\( 0 \)[/tex] and approximately [tex]\( 7.91 \)[/tex].

Given this information, the set of appropriate numbers for the domain of this function is Positive Real Numbers.