# Heating and Cooling

## Homework Statement

A solar hot-water-heating system consists of a hot water tank and a solar panel. The tank is well insulated and has a time constant of 64 hr. The solar panel generates 2000 Btu/hr during the day, and the tank has a heat capacity of 2 degrees F per thousand Btu. If the water in the tank is initially 110 F and the room temperature outside the tank is 80 degrees F, what will be the temperature in the tank after 12 hr of sunlight.

## Homework Equations

$$T(t) = e^{-kt} (\int e^{kt}[KM(t) + H(t) + U(t)]dt +C)$$

M is the outside temperature, H is other things that affect temperature in the tank(0 in this case), and U is the solar panel. K comes from the time constant, and should be the inverse of the time constant I believe. T is temperature, t is time.

## The Attempt at a Solution

$$T(t) = e^{-\frac{1}{64}t} (\int e^{\frac{1}{64}t}[ \frac{1}{64} (80) + 4t]dt$$

After integrating I keep getting

$$-16304 + 256t + Ce^{- \frac{1}{64} t}$$

I calculate C to be 16414 setting t equal to 0 and using the initial conditions. However, when plugging in 12 to my answer for t, I get in the 300s, whereas the book says the answer is 148.6. Could anyone please show me where I'm going wrong. I would greatly appreciate it.

Thanks for your time and help.