An underground gasoline tank can hold 1.07 103 gallons of gasoline at 52.0°F. If the tank is being filled on a day when the outdoor temperature (and the gasoline in a tanker truck) is 97.0°F, how many gallons from the truck can be poured into the tank? Assume the temperature of the gasoline quickly cools from 97.0°F to 52.0°F upon entering the tank. (The coefficient of volume expansion for gasoline is 9.6 10-4 (°C) - 1.)

1069.38 gallons

Explanation:

Let V₀ = 1.07 * 10³ be the initial volume of the gasoline at temperature θ₁ = 52 °F. Let V₁ be the volume at θ₂ = 97 °F.

V₁ = V₀ (1 + βΔθ) β = coefficient of volume expansion for gasoline = 9.6 * 10⁻⁴ °C⁻¹

Δθ = (5/9) (97°F - 52°F) °C = 25 °C.

Let V₂ be its final volume when it cools to 52°F in the tank is

V₂ = V₁ (1 - βΔθ) = V₀ (1 + βΔθ) (1 - βΔθ) = V₀ (1 - [βΔθ]²)

= 1.07 * 10³ (1 - [9.6 * 10⁻⁴ °C⁻¹ * 25 °C]²)

= 1.07 * 10³ (1 - [0.024]²)

= 1.07 * 10³ (1 - 0.000576)

= 1.07 * 10³ (0.999424)

= 1069.38 gallons