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math problem

Billy has 1 gallon of paint. He is going to pour it into a paint tray that measures 10 inches wide, 14 inches long, and 4 cm deep.

(1 gallon = 231 in3, 1 inch = 2.54 cm)

Which of the following scenarios will happen?

The paint will not fill the tray by 441 cm3.

The paint will not fill the tray by 10.53 in3.

The paint will fill the tray exactly.

The paint will overfill the tray by 10.53 in3.

Since 2 of 3 dimensions are in inches, there are 2  response choices choice with cubic inches versus 1 with cubic centimeters, 2 of three tray dimensions are in inches, and our gallon conversion factor yields cubic inches, it should be easiest if we start by converting the only metric dimension, the tray depth of 4 cm, to inches.  Since there are exactly 2.54 inches per cm, we start by dividing 4 by 2.54.

Here is a trick if you can’t remember whether to divide or multiply by your conversion factor, CF =2.54 cm/in.  You started with X cm and want Y inches   the answer is either

1) Y in =  X cm x  (CF in/cm)    or     2)  Y in = X cm / ( CF in /cm)     In the first  formula , the cm in the right half of the equation cancel leaving in, which is what you want. In the second formula, you end up with a denominator of squared cm, which is definitely not what you want.  Scientists and their students often apply this method called unit analysis to much more complex equations

In our case,  4cm /( 2.54 cm/in) =  1.5748 in. The number of significant figures is determined by the 4 cm, since 2.54 is an absolute constant. I don’t know to what accuracy the 4 cm height was measured, so let’s keep 3 digits for now.

The volume of the tray is just V = L x W x H   --   10 in x 14 in x  1.57 in  = 219.8 cubic in.

We are given that 1 gallon equals 231 cubic inches which overflows the tray.

Normally, we could stop here, because there is only one answer where the paint overflows the tray.  This kind of reasoning can save you time on standardized tests. However, since we are not working against the clock, let’s subtract the volume of the tray (219.8 cubic in ) from one gallon (231 cubic inches) leaving an overflow of 10.53 cubic inches, which confirms our choice.

Volume of paint tray = (10 inches) (14 inches) (4 cm) (1 inch)/(2.54 cm)
≈ 220.47 in^3.
A gallon is 231 in^3, which is more than the tray can hold by 231 - 220.47 = 10.53 in^3.
So the answer is: The paint will overfill the tray by 10.53 in^3.
paint tray volume is approximately 220.5 cubic inches.

231 - 220.5 = 10.5

one gallon is more that the tray can handle.

therefore, the paint will overfill the tray by 10.53 cubic inches.
Volume of tray is (14")*(10")*(4 cm) = (14")*(10")*[(4 cm)/(2.54 cm in-1) = 220.472 in3 or 3612.9 cm3

Gallon of paint is 231 in3 or 3785.41 cm3

The paint will overfill the tray by 3785.41 cm3 - 2612.9 cm3 = 172.512 cm3 = 10.5273 in3

Volume of the tray = 10 x 14 x (4/2.54) = 220.47 in3

1 gallon = 231 in3

So 1 gallon of piant is not going to fit in the tray

Overflow = 231 - 220.47 = 10.53 in3