93. Three start-ups Three cars, A, B, and C, start from rest and accelerate along a line according to the following velocity functions:
vₐ(t) = 88t/(t + 1), v_B(t) = 88t²/(t + 1)², and v_C(t) = 88t²/(t² + 1).
b. Which car travels farthest on the interval 0 ≤ t ≤ 5?

Verified step by step guidance

1. To determine which car travels the farthest on the interval \(0 \leq t \leq 5\), we need to find the total distance each car covers during this time. Since the velocity functions \(v_A(t)\), \(v_B(t)\), and \(v_C(t)\) are given, the distance traveled by each car is the integral of its velocity over the interval from 0 to 5.
2. Set up the definite integrals for each car's distance traveled: For car A: \[D_A = \int_0^5 \frac{88t}{t + 1} \, dt\] For car B: \[D_B = \int_0^5 \frac{88t^2}{(t + 1)^2} \, dt\] For car C: \[D_C = \int_0^5 \frac{88t^2}{t^2 + 1} \, dt\]
3. Next, evaluate each integral separately. This may involve techniques such as substitution or partial fraction decomposition depending on the integrand. For example, for \(D_A\), consider rewriting the integrand or using substitution \(u = t + 1\) to simplify the integral.
4. After finding the antiderivatives, apply the Fundamental Theorem of Calculus by evaluating each antiderivative at the upper limit \(t=5\) and subtracting the value at the lower limit \(t=0\) to find the exact distance traveled by each car.
5. Finally, compare the three distances \(D_A\), \(D_B\), and \(D_C\). The car corresponding to the greatest value among these integrals travels the farthest on the interval \(0 \leq t \leq 5\).

Key Concepts

Velocity and Displacement Relationship

Velocity is the rate of change of displacement with respect to time. To find the distance traveled over a time interval, you integrate the velocity function over that interval. This integral gives the total displacement, which is essential for comparing how far each car travels.

Definite Integration

Definite integration calculates the exact area under a curve between two points, representing accumulated quantities like displacement from velocity. Evaluating the integral of each velocity function from t = 0 to t = 5 will determine the total distance each car covers.

Analyzing Rational Functions

The given velocity functions are rational expressions involving polynomials in numerator and denominator. Understanding their behavior, such as growth rates and limits, helps anticipate which car might travel farther before performing integration.