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.
