Question

110. Comparing distances Suppose two cars started at the same time and place (t = 0 and s = 0). The velocity of car A (in mi/hr) is given by
u(t) = 40 / (t + 1) and the velocity of car B (in mi/hr) is given by v(t) = 40 * e^(-t/2).
b. After t = 3 hr, which car has traveled farther?

Accepted Answer

Recall that the distance traveled by a car from time $$t=0 to$$ $$t=3 is $$the integral of its velocity function over that interval. So, for each car, we need to compute $$\$$int_0$$^3 u(t) \, dt$$ and $$\$$int_0$$^3 v(t) \, dt$$ respectively. Set up the integral for car A's distance: $$\$$int_0$$^3 \frac{40}{t+1} \, dt. $$This integral involves a rational function and can be solved using the natural logarithm function. Set up the integral for car B's distance: $$\$$int_0$$^3 40 e$$^{-t/2}$$ \, dt. $$This integral involves an exponential function and can be solved using the formula for integrating exponentials. Evaluate both integrals separately by applying the appropriate integration techniques: for car A, use the substitution $$u = t+1$$; for car B, use the standard integral $$\int e$$^{kt}$$ dt = \frac{1}{k} e$$^{kt}$$ + C. $$After finding the expressions for the distances traveled by both cars at $$t=3$$, compare the two values to determine which car has traveled farther.

110. Comparing distances Suppose two cars started at the same time and place (t = 0 and s = 0). The velocity of car A (in mi/hr) is given by
u(t) = 40 / (t + 1) and the velocity of car B (in mi/hr) is given by v(t) = 40 * e^(-t/2).
b. After t = 3 hr, which car has traveled farther?

Verified video answer for a similar problem:

Key Concepts

Velocity and Displacement Relationship

Definite Integration

Exponential and Rational Functions

110. Comparing distances Suppose two cars started at the same time and place (t = 0 and s = 0). The velocity of car A (in mi/hr) is given byu(t) = 40 / (t + 1) and the velocity of car B (in mi/hr) is given by v(t) = 40 * e^(-t/2).b. After t = 3 hr, which car has traveled farther?

Verified video answer for a similar problem:

Key Concepts

Velocity and Displacement Relationship

Definite Integration

Exponential and Rational Functions

110. Comparing distances Suppose two cars started at the same time and place (t = 0 and s = 0). The velocity of car A (in mi/hr) is given by
u(t) = 40 / (t + 1) and the velocity of car B (in mi/hr) is given by v(t) = 40 * e^(-t/2).
b. After t = 3 hr, which car has traveled farther?