If a person was locked in a perfectly insulated room 8 ft by 8 ft by 10 ft how long would it take the room temperature to increase from 75 degrees F to 100 degrees F? A person's body heat output is 100 watts and their volume is 3 cubic feet. Also, it is given that the density of air is 1.2 kg/cubic meter and 1 watt sec/(g K) is its specific heat.

Energy to cause temperature change is computed by Q = mC(dT), with Q = change in energy, m = mass, C = specific heat, and dT = temperature change.

Find the volume of the air (remember to exclude the person), and then find the mass of that air, given as 1.2kg/m^3 = 42.384kg/ft^3 (units of volume need to match).

m = (8*8*10 - 3)*(42.384kg/ft^3)

Specific heat is given as 1 W*s/g/K = 1000 W*s/kg/K. (units of mass need to match)

Convert the temperatures to Kelvin to match the specific heat. (Tip: converting to Celsius is sufficient as you are finding a difference between two temperatures.) C = (5/9)(F-32)

T1 = 75F = ?C
T2 = 100F = ?C
dT = T2-T1 = ?

Now we can find Q, the change in energy.
Q = ?

We're given the power output of a person. P = Q/t, so t = Q/P. t = ?

I get that Q, the change in energy is 37498366.5. Using P = Q/t, with P = 100 watt, I get that t = 3749836.665 seconds. The solution is .835 hours. When I divide t by 3600, I don't get the correct answer. Any suggestions?

To answer this type of problem it is very important to include the units at each step.

Volume of box
8 ft x 8 ft x 10 ft = 640 cuft

volume of air = (640-3) cuft = 637 cuft

1 cubic metre = 35.3 cubic feet

volume of air = 637 cuft/35.3 cuft m^-3
= 18.04 m^3

so mass of air = 18.04 m^3 x 1.2 kg m^-3 = 21.65 kg

T1 = 75 degF or 23.9 degC
T2 = 100 degF or 37.8 degC
dT = T2-T1 = 13.9 degC or 13.9 K

specific heat = 1 W s g^-1 K^-1

so energy required to raise temperature by 13.9 K is

energy required is

1 W s g^-1 K^-1
x 21650 g x 13.9 K

= 300935 W s

so time required if power is 100 W is

300935 W s/100 W = 3009.35 s

or 0.836 h

which is close enough to you answer.

To calculate how long it would take for the room temperature to increase from 75 degrees F to 100 degrees F, we need to consider several factors.

1. Calculate the heat capacity of the room:
The heat capacity is the amount of heat energy required to raise the temperature of the room by one degree. Since the room is perfectly insulated, we need to take into account the specific heat of air, density of air, and the volume of the room.

The heat capacity (C) can be calculated using the formula:
C = (density of air) x (specific heat of air) x (volume of the room)

In this case, the density of air is given as 1.2 kg/cubic meter, and the specific heat of air is 1 watt sec/(g K).

Since the volume of the room is given as 8 ft x 8 ft x 10 ft, which is equivalent to 2.44 m x 2.44 m x 3.05 m, the volume is approximately 18.06 cubic meters.

Plugging in the values, we get:
C = 1.2 kg/cubic meter x 1 watt sec/(g K) x 18.06 cubic meters

2. Calculate the heat energy required to raise the temperature:
The heat energy required to raise the temperature of the room can be calculated using the formula:
Q = C x (change in temperature)

In this case, the change in temperature is from 75 degrees F to 100 degrees F, which is equivalent to (100 - 75) degrees F = 25 degrees F. However, we need to convert this to Kelvin as the specific heat is given in Kelvin units.

From the conversion formula F = (9/5)C + 32, we can convert 25 degrees F to Celsius and then to Kelvin.

3. Calculate the time required:
The time required can be calculated using the formula:
Q = P x t

In this case, the heat energy (Q) required is obtained from the previous step. The person's body heat output (P) is given as 100 watts.

Rearranging the formula, we get:
t = Q / P

By following these steps and plugging in the given values, you can calculate the time it would take for the room temperature to increase from 75 degrees F to 100 degrees F.