A skier wears a jacket filled with goose down that is 15mm thick. Another skier wears a wool sweater that is 7.0mm thick. Both have the same surface area. Assuming the temperature difference between the inner and outer surfaces of each garment is the same, calculate the ratio(wool/goose down) of the heat lost due to conduction during the same time interval.
To calculate the ratio of the heat lost due to conduction for the wool sweater and the goose down jacket, you need to use the formula:
Q = (k * A * ΔT) / d,
where:
Q is the heat conducted,
k is the thermal conductivity of the material,
A is the surface area of the garment,
ΔT is the temperature difference (inner surface - outer surface), and
d is the thickness of the garment.
For each garment, we will calculate the heat conducted during the same time interval and then find the ratio.
Let's assume:
k for goose down jacket = 0.025 W/(m·K) (thermal conductivity of goose down),
k for wool sweater = 0.03 W/(m·K) (thermal conductivity of wool),
A = same for both garments (since they have the same surface area),
ΔT = same for both garments (assuming they have the same temperature difference),
d1 = 15 mm (thickness of goose down jacket),
d2 = 7.0 mm (thickness of wool sweater).
Now, let's calculate the ratio:
Using the formula Q = (k * A * ΔT) / d:
For the goose down jacket, Q1 = (0.025 * A * ΔT) / 15,
For the wool sweater, Q2 = (0.03 * A * ΔT) / 7.0.
To find the ratio of heat lost:
Ratio = Q2 / Q1 = ((0.03 * A * ΔT) / 7.0) / ((0.025 * A * ΔT) / 15),
Simplifying the equation, we can cancel out A and ΔT:
Ratio = (0.03 / 7.0) / (0.025 / 15),
Ratio = (0.03 * 15) / (7.0 * 0.025),
Solving the equation:
Ratio = 1.2857.
So, the ratio of the heat lost due to conduction for the wool sweater (wool/goose down) is approximately 1.2857.