Q 4 - Calculus

Question

Find your average velocity from the oak tree to the

stop sign anotherway, as follows: Differentiate the formula

for position to get a formula v(t) for your velocity,

in feet per second, t seconds after hitting the brakes.

Then use a definite integral to find the average velocity

during the time that you were trying to stop the car.

(Your final answer should of course be the same as the

average velocity you found in the last part)

Step-by-Step Solution

Answer

-0.74 feet per second

Step 1: Given

The function, gives the distance from the stop sign (in feet) t seconds after stepping on the brakes is 

s(t) = 3t³ - 12t² -9t + 54

Step 2: By differentiation method

$$\begin{gathered} s\left(t\right)=3t^3-12t^2-9t+54 \\ \frac{ds\left(t\right)}{dt}=9t^2-24t-9 \\ v\left(t\right)=9t^2-24t-9 \\ {\int }_0^{v}v\left(t\right)dt={\int }_0^{4.6}\left(9t^2-24t-9\right)dt \\ {\int }_0^{v}v\left(t\right)dt={\left[9\frac{t^3}{3}\right]}_0^{4.6}-{\left[24\frac{t^2}{2}\right]}_0^{4.6}-{\left[9\frac{t}{1}\right]}_0^{4.6} \\ v\left(t\right)={\left[9\frac{t^3}{3}\right]}_0^{4.6}-{\left[24\frac{t^2}{2}\right]}_0^{4.6}-{\left[9\frac{t}{1}\right]}_0^{4.6} \\ v\left(t\right)=292-254-41.4 \\ v\left(t\right)=-3.4 \\ \mathrm{This} \mathrm{is} \mathrm{the} \mathrm{velocity} \mathrm{on} \mathrm{the} \mathrm{time} \mathrm{interval} [0,4.6]. \\ \mathrm{Hence}, \mathrm{the} \mathrm{average} \mathrm{can} \mathrm{be} \mathrm{calculated} \mathrm{by} \frac{-3.4}{4.6}. \\ \mathrm{As} \mathrm{a} \mathrm{result}, \mathrm{the} \mathrm{average} \mathrm{speed} \mathrm{is} -0.74 \mathrm{feet} \mathrm{per} \mathrm{second}. \end{gathered}$$