Question

A volleyball is served from a height of 6 feet with an initial velocity of 33 feet per second. The situation is modeled using the function h = -6t^2 + 33t + 6, where h is the height in feet and t is time in seconds. Which of the following uses the function to correctly identify the corresponding y-values to the given x-values of 0, 1, 2, 3, and 4?
A. \((0, 0), (1, 27), (2, 42), (3, 45), (4, 36)\)

B. \((0, 6), (1, 33), (2, 48), (3, 51), (4, 42)\)

C. \((6, 0), (33, 1), (48, 2), (51, 3), (42, 4)\)

D. \((0, 6), (1, 21), (2, 48), (3, 69), (4, 90)\)

Answer 1

To identify the corresponding y-values for the given x-values using the function \( h = -6t^2 + 33t + 6 \), we need to calculate the height \( h \) for each time \( t \) (which corresponds to the x-values: 0, 1, 2, 3, and 4).

Let's compute the height for each time value:

1. For \( t = 0 \):
\[
h(0) = -6(0^2) + 33(0) + 6 = 6
\]

2. For \( t = 1 \):
\[
h(1) = -6(1^2) + 33(1) + 6 = -6 + 33 + 6 = 33
\]

3. For \( t = 2 \):
\[
h(2) = -6(2^2) + 33(2) + 6 = -6(4) + 66 + 6 = -24 + 66 + 6 = 48
\]

4. For \( t = 3 \):
\[
h(3) = -6(3^2) + 33(3) + 6 = -6(9) + 99 + 6 = -54 + 99 + 6 = 51
\]

5. For \( t = 4 \):
\[
h(4) = -6(4^2) + 33(4) + 6 = -6(16) + 132 + 6 = -96 + 132 + 6 = 42
\]

Now we can summarize the values we calculated:
- \( t = 0 \rightarrow h = 6 \)
- \( t = 1 \rightarrow h = 33 \)
- \( t = 2 \rightarrow h = 48 \)
- \( t = 3 \rightarrow h = 51 \)
- \( t = 4 \rightarrow h = 42 \)

Putting them together, we have the points:
\[
(0, 6), (1, 33), (2, 48), (3, 51), (4, 42)
\]

Now, comparing these points with the options given:

A. \((0, 0), (1, 27), (2, 42), (3, 45), (4, 36)\)

B. \((0, 6), (1, 33), (2, 48), (3, 51), (4, 42)\)

C. \((6, 0), (33, 1), (48, 2), (51, 3), (42, 4)\)

D. \((0, 6), (1, 21), (2, 48), (3, 69), (4, 90)\)

The correct option is:

**B. \((0, 6), (1, 33), (2, 48), (3, 51), (4, 42)\)**