Mathematics

Problem

A laboratory technician heats a liquid whose temperature (in degrees Celsius) after

t

minutes is modeled by

T(t)=3t^{2}+mt+20.

The technician wants the average rate of change of the temperature over the first four minutes to be exactly

26\,^{\circ}\!\text{C}\text{/min}

. a. Derive an expression, in terms of

m

, for the average rate of change of

T

over the interval

[0,4]

. State your result in degrees Celsius per minute. b. Determine the value of

m

that yields the desired average rate of change. c. Interpret the meaning of your value of

m

in the context of the heating process.

Intuitive explanation

The average rate of change of a function over an interval is the slope of the secant line connecting its endpoints—essentially how much the output changes divided by how much time passes. Here, you need to find the value of $m$ that makes this average rate exactly $26\,^{\circ}\text{C/min}$ from $t=0$ to $t=4$ . Set up the average rate formula $\frac{T(4)-T(0)}{4-0}$ , substitute the given model, and solve for $m$ .

Intuitive explanation

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