Let's solve the problem step-by-step to find the best approximation for the height of the ball after the 3rd bounce.
1. Understand the given equation: The height [tex]\( y \)[/tex] of the ball after [tex]\( x \)[/tex] bounces is modeled by the quadratic equation:
[tex]\[
y = -3.5x^2 - 0.5x + 65
\][/tex]
2. Determine what we need to find: We need to calculate the height of the ball after the 3rd bounce. Hence, we need to substitute [tex]\( x = 3 \)[/tex] into the equation.
3. Substitute [tex]\( x = 3 \)[/tex] into the equation:
[tex]\[
y = -3.5(3)^2 - 0.5(3) + 65
\][/tex]
4. Perform the calculations:
- First, calculate [tex]\( 3^2 \)[/tex]:
[tex]\[
3^2 = 9
\][/tex]
- Next, calculate [tex]\( -3.5 \times 9 \)[/tex]:
[tex]\[
-3.5 \times 9 = -31.5
\][/tex]
- Then, calculate [tex]\( -0.5 \times 3 \)[/tex]:
[tex]\[
-0.5 \times 3 = -1.5
\][/tex]
- Now, add these results to the constant term [tex]\( 65 \)[/tex]:
[tex]\[
y = -31.5 - 1.5 + 65 = -33 + 65 = 32
\][/tex]
5. Determine the closest option: We have the calculated height after the 3rd bounce as 32 inches. Now, compare with the given options:
- 7 inches
- 14 inches
- 32 inches
- 50 inches
The best approximation from the given options is 32 inches.
Therefore, the height of the ball after the 3rd bounce is best approximated to be 32 inches.
