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Math  /  Calculus

QuestionCurrent Attempt in Progress The velocity of a car is f(t)=4tf(t)=4 t meters /sec/ \mathrm{sec}. Use a graph of f(t)f(t) to find the exact distance traveled by the car, in meters, from t=0t=0 to t=6t=6 seconds.
The distance traveled by the car is □\square meters. eTextbook and Media

Studdy Solution

STEP 1

1. The velocity function of the car is given by f(t)=4t f(t) = 4t meters/second.
2. The time interval for which the distance is to be calculated is from t=0 t = 0 to t=6 t = 6 seconds.
3. The distance traveled is the area under the velocity-time graph from t=0 t = 0 to t=6 t = 6 .

STEP 2

1. Understand the relationship between velocity and distance.
2. Set up the integral to calculate the area under the curve.
3. Evaluate the integral to find the exact distance.

STEP 3

Understand that the distance traveled by the car is the integral of the velocity function over the given time interval. This is equivalent to finding the area under the curve of f(t)=4t f(t) = 4t from t=0 t = 0 to t=6 t = 6 .

STEP 4

Set up the integral to calculate the area under the curve:
Distance=∫06f(t) dt=∫064t dt \text{Distance} = \int_{0}^{6} f(t) \, dt = \int_{0}^{6} 4t \, dt

STEP 5

Evaluate the integral:
∫4t dt=2t2+C \int 4t \, dt = 2t^2 + C
Now, calculate the definite integral from t=0 t = 0 to t=6 t = 6 :
∫064t dt=[2t2]06 \int_{0}^{6} 4t \, dt = \left[ 2t^2 \right]_{0}^{6}
=2(6)2−2(0)2 = 2(6)^2 - 2(0)^2
=2(36)−0 = 2(36) - 0
=72 = 72
The exact distance traveled by the car is:
72 meters \boxed{72} \text{ meters}

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