Day hike The velocity (in mi/hr) of a hiker walking along a straight trail is given by v(t) = 3 sin² πt/2, for 0≤t≤4. Assume s(0)=0 and t is measured in hours.

c. What is the hiker’s position at t=3?

검증된 단계별 안내

Step 1: Understand the problem. The hiker's position at time t=3 can be found by integrating the velocity function v(t) = 3 sin²(πt/2) over the interval [0, 3]. This is because position is the integral of velocity with respect to time.

Step 2: Set up the integral. The position function s(t) is given by s(t) = ∫ v(t) dt. Since s(0) = 0, the position at t=3 is s(3) = ∫[0 to 3] 3 sin²(πt/2) dt.

Step 3: Simplify the integrand using a trigonometric identity. Recall that sin²(x) can be rewritten as (1 - cos(2x))/2. Substituting this into the integral, we get s(3) = ∫[0 to 3] 3 * (1 - cos(πt))/2 dt.

Step 4: Break the integral into two parts. The integral becomes s(3) = (3/2) ∫[0 to 3] 1 dt - (3/2) ∫[0 to 3] cos(πt) dt. Evaluate each part separately.

Step 5: Evaluate the integrals. For the first term, ∫[0 to 3] 1 dt is straightforward and equals t evaluated from 0 to 3. For the second term, ∫[0 to 3] cos(πt) dt requires substitution u = πt, du = π dt, and then solving the integral. Combine the results to find s(3).

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Velocity and Position Functions

In calculus, velocity is the rate of change of position with respect to time. The position function, s(t), can be derived from the velocity function, v(t), by integrating it over time. In this case, the hiker's position at any time t can be found by integrating the given velocity function v(t) = 3 sin²(πt/2) from 0 to t.

Integration

Integration is a fundamental concept in calculus that allows us to find the accumulated value of a function over an interval. To find the hiker's position at t=3, we need to compute the definite integral of the velocity function from 0 to 3. This process essentially sums up all the small changes in position over the specified time period.

Definite Integral

A definite integral calculates the net area under a curve between two points on the x-axis. In this context, it represents the total distance traveled by the hiker from time t=0 to t=3. The result of the definite integral will give us the hiker's position at t=3, which is crucial for answering the question.

Day hike The velocity (in mi/hr) of a hiker walking along a straight trail is given by v(t) = 3 sin² πt/2, for 0≤t≤4. Assume s(0)=0 and t is measured in hours. c. What is the hiker’s position at t=3?