Question

The velocity of a car over five hours is given by v(t) = 60 ln(t + 1), 0 < t < 5

in kilometers per hour. What is the total distance traveled from t = 0 to t = 5?
Round your answer to the nearest whole number and do not include units.

Answer 1

345

Explanation:

The velocity of a car over five hours is given by:

`\displaystyle v(t)=60\ln(t+1),\, 0 \leq t \leq 5`

And we want to find the total distance traveled from t = 0 to t = 5.

Recall that distance is the integral of the absolute value of the velocity function. Since we want to find the total distance traveled from t = 0 to t = 5, our limits of integration are t = 0 and t = 5. Hence:

`\displaystyle D=\int_0^5 |\underbrace{60\ln(t+1)}_(v(t))|\, dt`

Since v(t) ≥ 0 for all t in the interval [1, 5], we can remove the absolute value. Use a calculator:

`\displaystyle D=60\left(6\ln(6)-5)=345.0334...\approx 345`