Question

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?

Accepted Answer

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. 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: 
$$D_C = \int_0^5 \frac{88t^2}{t^2 + 1} \, dt$$ 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. 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. 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 $$0 \leq t \leq 5.$$

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?

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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?

Risposta video verificata per un problema simile:

Concetti chiave

Velocity and Displacement Relationship

Definite Integration

Analyzing Rational Functions

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?