The braking distance [tex]\( D(v) \)[/tex] (in meters) for a certain car moving at velocity [tex]\( v \)[/tex] (in meters/second) is given by [tex]\( D(v) = \frac{v^2}{26} \)[/tex].

The car's velocity [tex]\( B(t) \)[/tex] (in meters/second) [tex]\( t \)[/tex] seconds after starting is given by [tex]\( B(t) = 3t \)[/tex].

Write a formula for the braking distance [tex]\( S(t) \)[/tex] (in meters) after [tex]\( t \)[/tex] seconds.

It is not necessary to simplify.

[tex]\[ S(t) = \][/tex]

[tex]\[ \boxed{\quad} \][/tex]



Answer :

Certainly! Let's solve this step-by-step.

1. Start with the given formulas:
- The braking distance [tex]\( D(v) \)[/tex] is given by:
[tex]\[ D(v) = \frac{v^2}{26} \][/tex]
- The car's velocity [tex]\( B(t) \)[/tex] at time [tex]\( t \)[/tex] seconds is given by:
[tex]\[ B(t) = 3t \][/tex]

2. We need to find the braking distance [tex]\( S(t) \)[/tex] as a function of time [tex]\( t \)[/tex]. To do this, we will substitute the velocity function [tex]\( B(t) \)[/tex] into the braking distance function [tex]\( D(v) \)[/tex].

3. Substitute [tex]\( B(t) \)[/tex] into [tex]\( D(v) \)[/tex]:
- Since [tex]\( v = B(t) \)[/tex], we can rewrite [tex]\( v \)[/tex] in the braking distance formula:
[tex]\[ v = 3t \][/tex]
- Now, substitute [tex]\( v = 3t \)[/tex] into the braking distance formula [tex]\( D(v) \)[/tex]:
[tex]\[ D(3t) = \frac{(3t)^2}{26} \][/tex]

4. Simplify the expression:
[tex]\[ D(3t) = \frac{(3t)^2}{26} = \frac{9t^2}{26} \][/tex]

5. Therefore, the braking distance [tex]\( S(t) \)[/tex] as a function of time [tex]\( t \)[/tex] is:
[tex]\[ S(t) = \frac{9t^2}{26} \][/tex]

So the answer is:
[tex]\[ S(t) = \frac{9t^2}{26} \][/tex]