Cycling distance A cyclist rides down a long straight road with a velocity (in m/min) given by v(t) = 400−20t, for 0≤t≤10, where t is measured in minutes.

b. How far does the cyclist travel in the first 10 min?

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Identify the velocity function given: $v(t) = 400 - 20t$, where $t$ is in minutes and $v(t)$ is in meters per minute.

Recall that the distance traveled over a time interval can be found by integrating the velocity function over that interval. So, the distance $D$ traveled from $t=0$ to $t=10$ is given by the definite integral $D = \int_0^{10} v(t) \, dt$.

Set up the integral explicitly: $D = \int_0^{10} (400 - 20t) \, dt$.

Integrate the function term-by-term: the integral of $400$ with respect to $t$ is \$400t$, and the integral of $-20t$ with respect to $t$ is $-10t^2$.

Evaluate the resulting expression $400t - 10t^2$ at the upper limit $t=10$ and subtract the value at the lower limit $t=0$ to find the total distance traveled.

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주요 개념

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

Velocity as a Function of Time

Velocity describes the rate of change of position with respect to time. In this problem, velocity is given as a function v(t) = 400 - 20t, which means the cyclist's speed changes linearly over time from 400 m/min to 200 m/min during the first 10 minutes.

Definite Integral for Displacement

The total distance traveled over a time interval can be found by integrating the velocity function over that interval. The definite integral of v(t) from t=0 to t=10 gives the net displacement, representing how far the cyclist has traveled in those 10 minutes.

Limits of Integration and Time Interval

The problem specifies the time interval 0 ≤ t ≤ 10 minutes. Setting the correct limits of integration ensures the calculation covers the entire duration of interest, capturing the cyclist's motion from start to 10 minutes exactly.

Cycling distance A cyclist rides down a long straight road with a velocity (in m/min) given by v(t) = 400−20t, for 0≤t≤10, where t is measured in minutes.b. How far does the cyclist travel in the first 10 min?