Question

Use the table to answer the question.

Time (sec.) 0 0.2 0.6 1 1.5 2
Height (ft.) 0 4 12 18 20 16
A soccer ball was kicked, and the height reached by the ball was recorded several times over the first 2 seconds of its path. Find the quadratic function that represents the dataset. What is the value of b? Round the value to the nearest hundredth.

(1 point)
In the quadratic function that represents the dataset, b=
.

Answer 1

To find the quadratic function representing the height of the soccer ball in relation to time, we can use the general form of a quadratic function:

\[ h(t) = at^2 + bt + c \]

where \( h(t) \) is the height, \( t \) is time, \( a \), \( b \), and \( c \) are coefficients we need to determine.

From the data provided:

- **At \( t = 0 \) seconds, \( h(0) = 0 \) ft.**
- **At \( t = 0.2 \) seconds, \( h(0.2) = 4 \) ft.**
- **At \( t = 0.6 \) seconds, \( h(0.6) = 12 \) ft.**
- **At \( t = 1 \) second, \( h(1) = 18 \) ft.**
- **At \( t = 1.5 \) seconds, \( h(1.5) = 20 \) ft.**
- **At \( t = 2 \) seconds, \( h(2) = 16 \) ft.**

With the height \( h(0) = 0 \) at \( t = 0 \), this gives us the value of \( c = 0 \), so we can simplify our function to:

\[ h(t) = at^2 + bt \]

We can use the other points to set up a system of equations:

1. For \( t = 0.2 \):
\[
h(0.2) = a(0.2)^2 + b(0.2) = 4
\]
\[
0.04a + 0.2b = 4 \quad \text{(1)}
\]

2. For \( t = 0.6 \):
\[
h(0.6) = a(0.6)^2 + b(0.6) = 12
\]
\[
0.36a + 0.6b = 12 \quad \text{(2)}
\]

3. For \( t = 1 \):
\[
h(1) = a(1)^2 + b(1) = 18
\]
\[
a + b = 18 \quad \text{(3)}
\]

Now we have a system of three equations. We can solve equations (1), (2), and (3).

From equation (3), we can express \( b \) in terms of \( a \):

\[
b = 18 - a
\]

Substituting \( b \) into equations (1) and (2):

Substituting into equation (1):

\[
0.04a + 0.2(18 - a) = 4
\]
\[
0.04a + 3.6 - 0.2a = 4
\]
\[
-0.16a + 3.6 = 4
\]
\[
-0.16a = 0.4
\]
\[
a = -\frac{0.4}{0.16} = -2.5
\]

Now substituting \( a = -2.5 \) back into the equation for \( b \):

\[
b = 18 - (-2.5) = 18 + 2.5 = 20.5
\]

Thus, we have:

- \( a = -2.5 \)
- \( b = 20.5 \)
- \( c = 0 \)

Finally, rounding \( b = 20.5 \) to the nearest hundredth,

the value of \( b \) is:

\[
\boxed{20.50}
\]