Weird problem with cm cubed divided by cm squared?

Question

The problem is:
253.14 cm^3 / 7.58 cm^2

The book's answer key says the answer is 33.4 cm
My question is, how is this possible?? How do you convert from 3-D volume to 2-D area and end up with an answer that specifies neither?
Thanks in advance

Answer 1

You divide the units. cm^3 / cm^2 = cm.



For example, if you have a box and are given the volume in cubic cm, and the area of the base in square cm, then divide the two and you get the height in regular cm.

Answer 2

exponents on units, where you are multiplying or dividing, act like exponents on numbers

e.g.

cm^3 / cm^2 = cm^(3-2) = cm^1 = cm, just like

a^3 / a^2 = a^(3-2) = a^1 = a.



It is true that you cannot add or subtract quantities whose units are different.

No such problem for multiplication or division.



Read an introductory physics book, and you will see such unit manipulation everywhere.

Answer 3

Let's say you have a brick.

You lay the brick on a piece of paper and make its outline of one of the side with a pencil. You measure the area of that outline (a rectangle most probably as bricks are blocks).

And it turns out to have an area L×W = 7.58 cm^2.



Then you flip the brick and measure its height (i.e. length from one flat face to the other parallel to it). It turns out to have H = 33.4 cm.



So how would you figure out volume of the brick?

I hope you remember from somewhere that volume of a block (nerd: cuboid) = L×W×H = area * height.

So (7.58 cm^2) * (33.4 cm) = 253.14 cm^3 (not really but we're kinda close)

Volume is ofc measured in cm^3 because you're doing cm^2 * cm = cm*cm*cm = cm^3



So now let's say you forgot the height but you still have volume and base area written down. Then just rearrange equation.

(7.58 cm^2) * height = 253.14 cm^3

Divide with area

height = 253.14 cm^3 / (7.58 cm^2)



Units are just like multipliers, i.e. you can treat them like variables (x, y, a, f whatever). They cancel out.

When you divide cm^3 with cm^2, you get cm^(3-2) = cm^1 = length.



= 253.14 / 7.58 × cm^3 / cm^2 = 33.4 cm



Best wishes!

Answer 4

V = A h

253 • 14 = 7 • 58 h

h = 253 • 14 / 7 • 58 cm

h = 33 • 4 cm

Answer 5

A volume is an area stacked on itself along the height axis. Thus, if you divide volume by area, you get the length of that height. This is probably the best conceptual model to deal with the issue.



I do recognize that an area has no height at all, so there is a minor kink in the mental image. But the same applies for thinking of a line as a series of points. Points have no length, yet if you line them up, they gain one as a line.



So, when you convert between different orders (dimensions), you end up with the magnitude of the one (or more) missing orders - the one(s) needed to make the higher-order dimension from the lower-order dimension. That is simply how things work, or how we choose to look at things perhaps is a better way to say it.

Answer 6

253.14 cm^3 / 7.58 cm^2

= [(253.14) / (7.58)] * [(cm^3 / cm^2)]

= 33.14 (cm^(3 - 2))

= 33.4 cm (centimeters)

Answer 7

Posting the problem will help.

Maybe you are asked to calculate for a length given volume and cross section area.