Q. 1 TF - Calculus

Question

Just as you are driving past the big oak tree on Main Street, you notice a new

stop sign 54 feet ahead of you. You slam on your brakes and end up coming to a full stop exactly at the stop sign.

Suppose that your distance from the stop sign (in feet) t seconds after stepping on the brakes is given by the function$s\left(t\right) = 3t^3 -12t^2 -9t+54$. By working through the following five problems you will see another argument that the distance travelled is related to the signed area under the velocity curve; this set of problems previews the Second Fundamental Theorem of Calculus, which we will see in Section 4.7.

How long did it take you to come to a full stop?

Step-by-Step Solution

Answer

$4.6$seconds will take to come to a full stop.

Step 1. Given information

The given function for the distance from the stop sign is:

$s\left(t\right) = 3t^3 -12t^2 -9t+54$

Step 2. Time took to a full stop the car

A stop sign is noticed $54$feet before and the car stops at the sign.

Substitute $54$for the function value in the function $s\left(t\right)$and solve the equation for t.

$$\begin{gathered} s\left(t\right) = 3t^3 -12t^2 -9t+54 \\ 54 = 3t^3 -12t^2 -9t+54 \\ 3t^3 -12t^2 -9t=0 \\ 3t(t-4.6)(t+0.6)=0 \\ t=0,-0.6,4.6 \end{gathered}$$

as $t>0$

so $t=4.6$

$4.6$ seconds will take to stop the car.