Question

The height of a ball thrown upward from the ground is given by the quadratic function

h = 120t - 16t2 , where h is in feet and t is in seconds.

What is the reasonable range for this function?
Responses
A {y: 0 ≤ y ≤ 120}{y: 0 ≤ y ≤ 120}
B {y: 0 ≤ y ≤ 225}{y: 0 ≤ y ≤ 225}
C {y: y ≥ 0}{y: y ≥ 0}
D all real numbers

Answer 1

To determine the reasonable range for the height function \( h(t) = 120t - 16t^2 \), we need to analyze the quadratic function.

1. **Identify the function:** The function represents the height of the ball over time and is a downward-opening parabola because the coefficient of \( t^2 \) is negative.

2. **Find the vertex:** The maximum height (the vertex) can be found using the formula:
\[
t = -\frac{b}{2a}
\]
where \( a = -16 \) and \( b = 120 \). Thus,
\[
t = -\frac{120}{2 \times -16} = \frac{120}{32} = 3.75 \text{ seconds}
\]

3. **Calculate the maximum height:** Now substitute \( t = 3.75 \) back into the height function to find the maximum height:
\[
h(3.75) = 120(3.75) - 16(3.75^2)
\]
First, calculate \( 3.75^2 = 14.0625 \):
\[
h(3.75) = 120(3.75) - 16(14.0625)
\]
\[
h(3.75) = 450 - 225 = 225 \text{ feet}
\]

4. **Determine the reasonable range:** The ball starts at height \( h = 0 \) when \( t = 0 \) and reaches a maximum height of \( 225 \) feet at \( t = 3.75 \) seconds, after which it will fall back to height \( h = 0 \) when it hits the ground (as \( t \) increases to about \( 7.5 \) seconds).

Thus, the reasonable range for the height \( h \) is from \( 0 \) feet to \( 225 \) feet.

The correct response is:
**B** \(\{y: 0 \leq y \leq 225\}\)