Help with a volume expansion problem

[KTANNE]

Homework Statement

A 3.00-liter aluminum cylinder at 5.00°C is filled to the brim with gasoline at the same temperature. If the aluminum and gasoline are warmed to 53.0°C, how much gasoline spills out? [Hint: Be sure to account for the expansion of the container. Also, ignore the possibility of evaporation, and assume the volume coefficients are good to three digits.

Homework Equations

Coeff of Linear Expansion for Al : 24 X 10^-6
Coeff of Volume Expansion for gasoline: 9.6 X 10 ^-4
Volume Expansion Formula: ΔV = βVoΔT

The Attempt at a Solution

I calculated the volume expansion formula for each variable and came up this:
aluminum = (24 X 10^-6)(3.00L)(53.0 - 5.00)°C = .003456
gasoline = (9.6 X 10^-4)(3.00L)(53.0 - 5.00)°C = .13824

I then subtracted (.13824) - (.003456) = .134784 = .135
And this is the incorrect answer... The correct answer is 128 cm^3 so I don't know what I am doing incorrectly...

 

[KTANNE]

I figured it out - NM

 

[jk27]

how did you solve this problem?

 

[Redbelly98]

jk27, please stand by, I or somebody will post some help in your thread.

p.s. Welcome to PF to both of you.

 

[Nickie]

I would like to commend you for taking the time to work on this problem and seeking help when needed. It shows that you are dedicated to understanding and mastering the material.

Now, let's take a closer look at the problem and your attempt at a solution. From the given information, we can see that there are two main factors that contribute to the volume expansion: the expansion of the aluminum cylinder and the expansion of the gasoline.

First, let's calculate the volume expansion of the aluminum cylinder. Using the given coefficient of linear expansion for aluminum (24 x 10^-6), we can calculate the change in length of the cylinder as:

ΔL = (24 x 10^-6)(3.00L)(53.0 - 5.00)°C = 0.003456 m

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

ΔV = πr^2ΔL = 3.14(0.1525)^2(0.003456) = 0.000265 m^3

Next, let's calculate the volume expansion of the gasoline. Using the given coefficient of volume expansion for gasoline (9.6 x 10^-4), we can calculate the change in volume of the gasoline as:

ΔV = (9.6 x 10^-4)(3.00L)(53.0 - 5.00)°C = 0.1296 m^3

Now, we need to take into account the expansion of the container. Since the container is a cylinder, we can use the formula for the volume of a cylinder to calculate the change in volume:

ΔV = πr^2ΔL = 3.14(0.1525)^2(0.003456) = 0.000265 m^3

Adding the volume expansions of the aluminum cylinder, the gasoline, and the container, we get a total volume expansion of:

ΔVtotal = 0.000265 + 0.1296 + 0.000265 = 0.13013 m^3

Finally, we need to convert this volume to the desired unit of cm^3. Since 1 m^3 = 1,000,000 cm^3, we can multiply our total volume expansion by