Question
At t sec after liftoff, the height of a rocket is \( 5 t^{2} \mathrm{ft} \). How fast is the rocket The speed of the rocket 9 seconds after liftoff is (Simplify your answer.)
Ask by Nguyen Griffin. in the United States
Mar 09,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
The speed of the rocket 9 seconds after liftoff is 90 ft/s.
Solution
Given the height function
\[
h(t) = 5t^2,
\]
we first find the velocity by taking the derivative with respect to \( t \). This gives us
\[
v(t) = \frac{dh}{dt} = 10t.
\]
Next, we evaluate the velocity at \( t = 9 \) seconds:
\[
v(9) = 10 \times 9 = 90.
\]
Thus, the speed of the rocket 9 seconds after liftoff is
\[
90 \text{ ft/s}.
\]
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Simplify this solution Beyond the Answer
To find the speed of the rocket at \( t = 9 \) seconds, we need to determine the derivative of the height function \( h(t) = 5t^2 \) with respect to time \( t \). The derivative, \( h'(t) \), gives us the velocity. Calculating the derivative: \[ h'(t) = \frac{d}{dt}(5t^2) = 10t \] Now, we can find the speed at \( t = 9 \) seconds: \[ h'(9) = 10 \times 9 = 90 \text{ ft/sec} \] So, the speed of the rocket 9 seconds after liftoff is \( 90 \text{ ft/sec} \).
