# Question Video: Evaluating Piecewise-Defined Functions in a Real-World Context Mathematics

Given that the velocity of a motorcycle is expressed by the function 𝑣(𝑡) = 5𝑡, 0 ≤ 𝑡 ≤ 12, 𝑣(𝑡) = 60, 12 < 𝑡 < 70, 𝑣(𝑡) = −𝑡 + 130, 70 ≤ 𝑡 ≤ 130, where 𝑡 is the time in seconds, and 𝑣 is the velocity in m/s, determine 𝑣(82).

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### Video Transcript

Given that the velocity of a motorcycle is expressed by the function 𝑣 of 𝑡 is equal to five 𝑡 when 𝑡 is greater than or equal to zero and 𝑡 is less than or equal to 12 and 𝑣 of 𝑡 is equal to 60 when 𝑡 is greater than 12 and 𝑡 is less than 70 and 𝑣 of 𝑡 is equal to negative 𝑡 plus 130 when 𝑡 is greater than or equal to 70 and 𝑡 is less than or equal to 130, where 𝑡 is the time in seconds and 𝑣 is the velocity in meters per second, determine 𝑣 evaluated at 82.

In this question, we’re given a piecewise-defined function 𝑣 of 𝑡, which tells us the velocity of a motorcycle after 𝑡 seconds, measured in meters per second. We need to use this to determine the value of 𝑣 evaluated at 82. That’s the velocity of the motorcycle at 𝑡 is equal to 82 seconds. To evaluate 𝑣 at 82, we need to substitute 𝑡 is equal to 82 into our piecewise-defined function.

But remember, piecewise-defined functions are defined over different subdomains. So we first need to determine which of the subdomains our value of 82 lies in. And to do this, we need to determine which of the three inequalities our input value of 𝑡 satisfies. And of course we know 82 is greater than or equal to 70 and 82 is less than or equal to 130. So 82 lies in the third subdomain of our function 𝑣 of 𝑡. Therefore, when 𝑡 is equal to 82, 𝑣 of 𝑡 is equal to the function negative 𝑡 plus 130.

Therefore, to evaluate 𝑣 at 82, we need to substitute 82 into the function negative 𝑡 plus 130. We get 𝑣 of 82 is negative 82 plus 130. And if we evaluate this expression, we get 48. And we could leave our answer like this. However, remember, when 𝑡 is measured in seconds, 𝑣 of 𝑡 tells us the velocity of our motorcycle in meters per second. So we can give our answer the units meters per second. In other words, when 𝑡 is equal to 82 seconds, the velocity of our motorcycle is 48 meters per second.

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