Question

A rocket is launched upward. Its height, h of t in feet, after t seconds, is modeled by the function

h of t equals 80 t minus 16 t squared.


What is the domain of h of t?

Answer options with 5 options
A.
all real numbers greater than 0

B.
all real numbers greater than 0 and less than 5

C.
all real numbers greater than 0 and less than 16

D.
all real numbers greater than 0 and less than or equal to 5

E.
all real numbers greater than 0 and less than or equal to 16

Answer 1

To determine the domain of the function \( h(t) = 80t - 16t^2 \), we need to identify the range of \( t \) (time in seconds) for which the height \( h(t) \) is non-negative (i.e., the rocket is above ground).

First, we need to find when the height \( h(t) \) is equal to 0:

\[
80t - 16t^2 = 0
\]

Factoring the equation:

\[
16t(5 - t) = 0
\]

This gives us two solutions:

1. \( 16t = 0 \) → \( t = 0 \)
2. \( 5 - t = 0 \) → \( t = 5 \)

This means the height is zero at \( t = 0 \) seconds and \( t = 5 \) seconds. Between these two points, the rocket will be above the ground. Thus, the function \( h(t) \) is defined and non-negative for:

\[
0 \leq t \leq 5
\]

Since the problem specifies the domain in terms of real numbers greater than 0, we exclude \( t = 0 \) from the interval.

Therefore, the valid domain of \( h(t) \) is:

\[
0 < t \leq 5
\]

The answer option that correctly describes this domain is:

**D. all real numbers greater than 0 and less than or equal to 5**