Piecewise velocity The velocity of a (fast) automobile on a straight highway is given by the functionv(t)={3t if 0≤t\u003c2060 if 20≤t\u003c45240−4t if t≥45 v(t)= \begin{cases}3 t & \text { if } 0 \leq t\u003c20 \\ 60 & \text { if } 20 \leq t\u003c45 \\ 240-4 t & \text { if } t \geq 45\end{cases} v(t)=⎩⎨⎧3t60240−4t if 0≤t\u003c20 if 20≤t\u003c45 if t≥45 , where is measured in seconds and v has units of m/s. c. What is the distance traveled by the automobile in the first 60 s?

Ch. 6 - Applications of Integration

Briggs - Calculus: Early Transcendentals 3rd Edition

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Briggs 3rd Edition

Ch. 6 - Applications of Integration

Problem 6.1.27c

Chapter 6, Problem 6.1.27c

Piecewise velocity The velocity of a (fast) automobile on a straight highway is given by the function
$v(t)= $\begin{cases}$3 t & $\text$ { if } 0 $\leq$ t<20 \\ 60 & $\text$ { if } 20 $\leq$ t<45 \\ 240-4 t & $\text$ { if } t $\geq$ 45$\end{cases}$$
, where is measured in seconds and v has units of m/s.

c. What is the distance traveled by the automobile in the first 60 s?

Verified step by step guidance

Understand that the distance traveled by the automobile over a time interval is the integral of the velocity function over that interval. Since velocity is given piecewise, we will integrate each piece over its respective time interval and then sum the results.

Set up the integral for the first interval where $v(t) = 3t$ for $0 \leq t < 20$. The distance traveled in this interval is $\int_0^{20} 3t \, dt$.

Set up the integral for the second interval where $v(t) = 60$ for $20 \leq t < 45$. The distance traveled in this interval is $\int_{20}^{45} 60 \, dt$.

Set up the integral for the third interval where $v(t) = 240 - 4t$ for $t \geq 45$. Since we want the distance up to 60 seconds, integrate from 45 to 60: $\int_{45}^{60} (240 - 4t) \, dt$.

Calculate each integral separately and then add the three results together to find the total distance traveled in the first 60 seconds.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Piecewise Functions

A piecewise function is defined by different expressions over distinct intervals of the domain. Understanding how to interpret and evaluate each piece separately is essential, especially when calculating quantities like velocity that change behavior over time.

Distance from Velocity via Integration

Distance traveled is found by integrating the velocity function over time. For velocity given as v(t), the distance from time a to b is the integral of |v(t)| dt, which sums the total displacement considering the velocity's magnitude.

Handling Piecewise Integration

When integrating a piecewise function, split the integral at the interval boundaries and compute each part separately. Then sum these results to find the total value, ensuring correct limits and function expressions are used for each segment.

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