Question

A 14 -gal container is filled with gasoline. Neglect the change in volume of the container, and find how many gallons are lost if the temperature increases by \(27^{\circ} \mathrm{F}\). The volume expansion coefficient of gasoline is \(9.6 \cdot 10^{-4}{ }^{\circ} \mathrm{C}^{-1}\).

Short answer

Answer: Approximately 0.202 gallons are lost.

Step-by-step solution

Convert the temperature change from Fahrenheit to Celsius

We are given a temperature change of 27°F. We need to convert this to Celsius, using the following formula: 1°C = (5/9) °F So, a change of 27°F can be converted to Celsius as: Change in Celsius = (5/9) * 27 Now, we can calculate the change in Celsius: Change in Celsius = 15°C

Use the volume expansion coefficient of gasoline to calculate the volume change

We are given the volume expansion coefficient of gasoline (\(γ\)) as \(9.6 \times 10^{-4} °\mathrm{C}^{-1}\). We can use this value along with the temperature change in Celsius (ΔT) to calculate the volume change (ΔV) using the formula: ΔV = V_initial * γ * ΔT Where V_initial is the initial volume of gasoline in the container (14 gallons). Now, we can plug the values into the formula: ΔV = 14 gallons * (\(9.6 \times 10^{-4} °\mathrm{C}^{-1}\)) * 15°C

Calculate the volume of gasoline lost

To find the volume of gasoline lost, we first need to calculate the final volume of gasoline after the temperature increase: ΔV = 14 gallons * (\(9.6 \times 10^{-4} °\mathrm{C}^{-1}\)) * 15°C ΔV ≈ 0.202 gallons Now, to find the volume of gasoline lost, we take the difference between the initial and final volumes: Volume Lost = Initial Volume - Final Volume Volume Lost = 14 gallons - (14 gallons + 0.202 gallons) Volume Lost ≈ 0.202 gallons Thus, about 0.202 gallons of gasoline are lost when the temperature increases by 27°F.