Question

In the United States, building insulation is specified by the \(R\)-value (thermal resistance in $\mathrm{h} \cdot \mathrm{ft}^{2}+{ }^{\circ} \mathrm{F} /$ Btu units). A homeowner decides to save on the cost of heating the home by adding additional insulation in the attic. If the total \(R\)-value is increased from 15 to 25 , the homeowner can expect the heat loss through the ceiling to be reduced by (a) \(25 \%\) (b) \(40 \%\) (c) \(50 \%\) (d) \(60 \%\) (e) \(75 \%\)

Short answer

Answer: 40%

Step-by-step solution

Recall the relationship between R-value and heat loss

The \(R\)-value measures the thermal resistance of a material, which means that the higher the \(R\)-value, the better the insulation and the lower the heat loss. The heat loss is inversely proportional to the \(R\)-value: \(Q \propto \frac{1}{R}\), where \(Q\) is the heat loss and \(R\) is the \(R\)-value.

Calculate the initial heat loss

Let's denote the initial heat loss as \(Q_1\) and the final heat loss as \(Q_2\). Based on the initial \(R\)-value, we have \(Q_1 \propto \frac{1}{R_1}\), where \(R_1 = 15\).

Calculate the final heat loss

Similarly, based on the final \(R\)-value, we have \(Q_2 \propto \frac{1}{R_2}\), where \(R_2 = 25\).

Calculate the reduction in heat loss

To find the reduction in heat loss, we can compare the final heat loss (\(Q_2\)) to the initial heat loss (\(Q_1\)). The reduction in heat loss can be calculated as follows: Reduction in heat loss \(= \frac{Q_1 - Q_2}{Q_1} \times 100\%\) Since \(Q_1 \propto \frac{1}{R_1}\) and \(Q_2 \propto \frac{1}{R_2}\), we can rewrite the reduction in heat loss as: Reduction in heat loss \(= \frac{\frac{1}{R_1} - \frac{1}{R_2}}{\frac{1}{R_1}} \times 100\% = \frac{1 - \frac{R_1}{R_2}}{1} \times 100\%\) Now, we can plug in the given \(R\)-values (15 and 25) and calculate the reduction in heat loss: Reduction in heat loss \(= \frac{1 - \frac{15}{25}}{1} \times 100\% = \frac{10}{25} \times 100\% = 0.4 \times 100\% = 40\%\) So, the heat loss through the ceiling will be reduced by (b) \(40\%\).