Question
The position function of a golf ball rolling on an indine is given by \( d(t)=4 t^{2}+7 t \), where \( d \) is measured in meters, \( t \) in seconds. Find the ball's velocity at \( t=3 \) seconds.
Ask by Salazar Pearson. in the United States
Feb 20,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
The ball's velocity at 3 seconds is 31 meters per second.
Solution
The velocity of the golf ball is the derivative of the position function d(t) with respect to time. Given d(t) = 4t² + 7t, we differentiate:
d'(t) = 8t + 7
Substituting t = 3 seconds into the derivative:
d'(3) = 8(3) + 7 = 24 + 7 = 31
Thus, the ball's velocity at t = 3 seconds is 31 meters per second.
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Simplify this solution Beyond the Answer
To find the ball's velocity at \( t=3 \) seconds, we need to determine the derivative of the position function \( d(t) \). The position function is given by: \[ d(t) = 4t^{2} + 7t \] The velocity \( v(t) \) is the derivative of the position function: \[ v(t) = \frac{dd(t)}{dt} = \frac{d}{dt}(4t^{2} + 7t) \] Calculating the derivative: \[ v(t) = 8t + 7 \] Now, substituting \( t = 3 \) seconds into the velocity function: \[ v(3) = 8(3) + 7 = 24 + 7 = 31 \] Therefore, the ball's velocity at \( t=3 \) seconds is \( 31 \) meters per second.
