Finding Surface Area in square feet with Volume & Thickness

[Oannes]

Homework Statement

How large a surface area in units of square feet will 1 gallon of paint cover if we apply a coat of paint that is 0.1cm thick?

Homework Equations

Since Volume is L * W * H and we can assume the object is square besides the height which in this case will be the thickness. So since Volume = area * thickness then we can say that the area is Volume/Thickness. So:

Surface Area = Volume / Thickness
x ft² = 1gallon / 0.1 cm

The Attempt at a Solution

First I converted gallons to cubic feet which would give me the volume. There is .264 gallons in a liter, and .0353 cubic feet in a Liter. Therefore:

(1gal)(1L/.264gal)(.0353ft³/1L) = 0.1337ft³

Second I converted .1cm to feet, this was much easier:

(.1cm)(1m/100cm)(3.28ft/1m) = .00328ft

Thirdly I plugged them into the equation for finding Surface Area:

SA = (.1337ft³/.00328ft)

Resulting in:

SA = 40.76ft²

My Question: I feel that I worked this problem correctly but I am not sure and want to be positive that all the steps I followed are correct. My biggest issue is wondering if I can divide cubed feet by ft resulting in square feet. I believe this to be possible for you can do this with variables with exponents power of 3 in the numerator and to the power of 1 in the denominator, but I just want to be sure. If something is wrong I would appreciate any help in guiding me in the right direction. Thanks in advance for any help given, this problem racked my brain for a while until I started to think about what volume truly was in respect to dimensions.

 

[Bystander]

divide cubed feet by ft resulting in square feet

Perfectly legal.

 

[LCKurtz]

@Oannes: What you did is fine. Here's a suggestion on how you might organize it that you might like. You started with$$
x~ \text{ft}^2 = \frac {1 \text{gal}}{0.1 \text{cm}}$$Now you can just multiply by one conversion fraction after another like this:$$
x~ \text{ft}^2 = \frac {1 \text{gallon}}{0.1 \text{cm}} \cdot \frac{.133681\text{ft}^3}{1~\text{gallon}}
\cdot\frac{12\text{in}}{1~\text{ft}}\cdot \frac{2.54\text{cm}}{1~\text{inch}}=40.745\text{ft}^2$$Each conversion fraction changes the units, cancelling the old and putting in the new units. I did a slightly different string of conversions, but that's the nice thing about the method. Do it in any order you like, changing the units along the way.

 

[Oannes]

@Oannes: What you did is fine. Here's a suggestion on how you might organize it that you might like. You started with$$
x~ \text{ft}^2 = \frac {1 \text{gal}}{0.1 \text{cm}}$$Now you can just multiply by one conversion fraction after another like this:$$
x~ \text{ft}^2 = \frac {1 \text{gallon}}{0.1 \text{cm}} \cdot \frac{.133681\text{ft}^3}{1~\text{gallon}}
\cdot\frac{12\text{in}}{1~\text{ft}}\cdot \frac{2.54\text{cm}}{1~\text{inch}}=40.745\text{ft}^2$$Each conversion fraction changes the units, cancelling the old and putting in the new units. I did a slightly different string of conversions, but that's the nice thing about the method. Do it in any order you like, changing the units along the way.

Thanks for the help with cleaning up my problem, I tend to be messy and as a result of that I sometimes lose where I am! Thanks so much!

 

[Oannes]

@Bystander thanks a bunch I was sweating it! lol