Coefficient of Volume Expansion Question

[laurenh19]

Please Help! Coefficient of Volume Expansion Question

On a hot day (30⁰C) an oil trucker loaded with 50 000L of gasoline into his truck. By the time he got to his destination the temperature has dropped to -5⁰C. How many litres of fuel did he deliver? The coefficient of volume expansion of gasoline is 9.5 X10^-4 /⁰C. The coefficient of linear expansion for his steel tank is 11X10^-6/⁰C. The tank is a cylinder 10m long with a radius of 2m
V=V₀(1+3alpha $$\Delta$$T
$$\Delta$$V/V₀=$$\beta$$$$\Delta$$T
i figured out the volume of the cylinder is 40pi m³ but i don't know where to go from there.

 

[jbsteele]

The volume of the gasoline in the truck at -50C is given by: V = V0 (1 + 3αΔT)where V0 is the initial volume at 300C, α is the coefficient of volume expansion for gasoline and ΔT is the temperature change from 300C to -50C. In addition, the volume of the tank expands due to the temperature change, so the final volume of the gasoline delivered is given by:V = V0(1 + 3αΔT) - βΔTwhere β is the coefficient of linear expansion for the tank. Substituting the values, we get:V = 50000L (1 + 9.5 x 10-4/K × (-350K)) - 11 x 10-6/K × (-350K)= 50000L - 0.3325L= 49966.75L Therefore, the trucker delivered 49966.75L of gasoline.

 

[nlightner]

To answer this question, we need to use the formula for the coefficient of volume expansion:

ΔV/V0 = βΔT

We know the coefficient of volume expansion for gasoline is 9.5 x 10^-4 /°C and the temperature change is 350°C (300°C to -50°C). We also know the initial volume of gasoline is 50,000L, which is equivalent to 50m^3.

Using the formula, we can calculate the change in volume of the gasoline:

ΔV/50 = (9.5 x 10^-4 /°C)(350°C)

ΔV = 0.16625m^3

Now, we need to take into account the expansion of the steel tank. We can use the formula for the coefficient of linear expansion to calculate the change in length of the tank:

ΔL = (11 x 10^-6 /°C)(10m)(350°C)

ΔL = 0.385m

Since the tank is a cylinder, the change in radius will be half of the change in length, so Δr = 0.1925m.

Now, we can use the formula for the volume of a cylinder to calculate the change in volume of the tank:

ΔV = π(2m+0.1925m)^2(10m+0.385m) - π(2m)^2(10m)

ΔV = 0.1161m^3

Finally, we can calculate the total change in volume of the gasoline and tank:

Total ΔV = 0.16625m^3 + 0.1161m^3 = 0.28235m^3

To find the final volume of the gasoline, we need to add this change in volume to the initial volume:

Final volume = 50m^3 + 0.28235m^3 = 50.28235m^3

Converting this to litres, we get 50,282.35L of gasoline delivered by the trucker.

In summary, the trucker delivered 50,282.35L of gasoline to his destination.