The temperature (in degrees Fahrenheit) of water circulating through an industrial…

MATHEMATICS · Multi-part · AP · Calculus AB · Applications of Integration · Applications of Integration

Problem

The temperature (in degrees Fahrenheit) of water circulating through an industrial cooling loop is modeled for

0\le t\le 12

hours by

T(t)=A\sin\!\left(\dfrac{\pi t}{12}\right)+68.

Engineers want the average water temperature over this 12-hour period to be exactly

70^{\circ}\text{F}

. (a) Write, but do not evaluate, an expression whose value equals the average temperature of the water during the interval

0\le t\le 12

. Clearly indicate any integral limits. (b) Determine the value of the constant

A

that will achieve the desired average temperature of

70^{\circ}\text{F}

. Show all steps leading to your result. (c) Under practical operating conditions the instantaneous temperature must stay between

62^{\circ}\text{F}

and

78^{\circ}\text{F}

. Verify that the value of

A

you found in part (b) satisfies this operational constraint.