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.)

Answers

Answer 1

Answer:

Answer:

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

Answer 2

Answer:

Final answer:

To calculate the amount of gasoline that can be poured into the tank, we need to find the change in volume of the gasoline when its temperature changes from 97.0°F to 52.0°F. Using the equation for volume expansion, we can calculate this change in volume to be approximately 258 gallons.

Explanation:

To calculate the amount of gasoline that can be poured into the tank, we need to find the change in volume of the gasoline when its temperature changes from 97.0°F to 52.0°F. We can use the equation for volume expansion to calculate this change in volume:

ΔV = V₀ * β * ΔT

Where ΔV is the change in volume, V₀ is the initial volume, β is the coefficient of volume expansion, and ΔT is the change in temperature.

In this case, the initial volume V₀ is 1.07 * 10³ gallons, the coefficient of volume expansion β is 9.6 * 10⁻⁴ (°C)⁻¹, and the change in temperature ΔT is (52.0°F - 97.0°F) = -45.0°F.

Converting the change in temperature to Celsius: ΔT = (45.0°F) * (5/9) = -25.0°C.

Plugging in these values into the equation, we get:

ΔV = 1.07 * 10³ * 9.6 * 10⁻⁴ * -25.0 = -258 gallons.

Therefore, when the gasoline is poured into the tank, approximately 258 gallons will be poured out of the truck.

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