Answer to Question #137414 in Calculus for Nick

Question

The number of cars passing a motorway at any time t hours is given by the

N = 1/3t^3-5/5t^2-6t+980

find the time of which maximum number of cars passing the motorway and hence find the number of cars passing at that time.

Answer 1

$N = \frac{1}{3}t^3 - \frac{5}{2}t^2 -6t +980$

Find first derivative:

$N^\prime= t^2 -5t-6$

Then

$t^2 -5t -6=0\\ t_1 + t_2 = 5\\ t_1*t_2 =-6\\ t_1 = 6\\ t_2 = -1$

Find second derivative:

$N^{\prime\prime} = 2t -5$

And find $N(-1):$

$N(-1) = 2*(-1) - 7 < 0$ , so $t= -1$ - maximum

But time is supposed to be nonnegative, so choose the non-negative point closest to the maximum - 0.

And number of cars at that time:

$N(0) = 980$