Question

A car’s path is modeled by the function

y
⁢
(
t
)
=
−
4
t
2
+
24
⁢
t
+
8
, where y(t) represents the car’s position in meters along a parabolic trajectory after t seconds. How far will the car be at t=5 seconds? Find y(5).

Answer 1

To find the position of the car at \( t = 5 \) seconds, we need to evaluate the function \( y(t) \) at \( t = 5 \):

\[
y(t) = -4t^2 + 24t + 8
\]

Now substituting \( t = 5 \):

\[
y(5) = -4(5)^2 + 24(5) + 8
\]

Calculating each part step by step:

1. Calculate \( (5)^2 \):
\[
(5)^2 = 25
\]

2. Multiply by -4:
\[
-4 \times 25 = -100
\]

3. Calculate \( 24 \times 5 \):
\[
24 \times 5 = 120
\]

4. Now substitute these values back into the equation:
\[
y(5) = -100 + 120 + 8
\]

5. Combine the numbers:
\[
-100 + 120 = 20
\]
\[
20 + 8 = 28
\]

Thus, the position of the car at \( t = 5 \) seconds is

\[
\boxed{28} \text{ meters}.
\]