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### Course: High school geometry>Unit 9

Lesson 3: Volume and surface area

# Applying volume of solids

Calculate the volume of a grain hopper and then use the given rate to solve an applied problem. Created by Sal Khan.

## Want to join the conversation?

• Can't we also find the radius of the smaller cone by using the fact that the cones are similar? That is, if they are similar? Instead of taking a cross section and proving that the triangles in the cross section are similar... Is there any situation where this won't work?
• Sal does that at when he sets up the equation r/10=6/8 to find r= 7.5. Since it is a single cone, each measure will be proportional
• when u have a can of beans and the height is 11 and the rades is 12 what will be an answer. my mom really needs to know because we r going to be using some beans tonight with dinner. yummmmmmmmmmmmmmy. also we r having hot dog wennies but I will save that problem for later.
• What can of beans has a diameter that is more than twice the height? V=πr^2 h
• At about in the video, Sal multiplies 6 by 1/3. Wouldn't he have to do the same for the 7.5? Why doesn't he?

At , he says that dividing by 1/10 is the same as multiplying by 10. How is this possible? Also, why does he multiply by 10?
(1 vote)
• Multiplication is commutative, so we can move things around to make it easy to multiply. So 1/3*(7.5)^2*6 can be commuted to (7.5)^2 * 1/3 * 6. You are trying to apply the distributive property which is not the same, you are trying to do 1/3((7.5)^2 + 6) where both terms would be multiplied by 1/3, but the problem has a single term.
As to the second issue, dividing by a tenth can be shown in two ways, 353.4/.1, multiplying both numerator and denominator by 10 (10/10=1) gives 3534/1, and dividing by 1 does not change a number, so it is not needed. The second way is 353.4/(1/10). To divide a whole number by a fraction, we reciprocate the fraction and multiply which gives 353.4*(10/1)=3534.
• At , what triangles is he talking about
?
• he's talking about the triangles on the cone, basically, he cut a image of a cone into two right similar triangles
• can someone tell me why he puts the r on top for the proportion @ ?
why doesn't it go on the bottom?

I figure he's comparing the r to 10 so r:10 or r/10, but I need a deeper understanding as to why ?

Also, trying to conceptiualize if the ratio was 10/r in this problem @ how would the problem , the triangles, the question being asked, and answer differ?

not sure what im missing here, thanks
• i think his decision to put the r on top was mostly arbitrary

if the ratio was 10/r, then it would just be set up as 10/r = 8/6 instead of r/10 = 6/8. it would still come out to 7.5 and the rest of the problem would be the same

hope this helps!
(1 vote)
• from my proportion I think of it like 10/8=r/6 or 8/10=6/r am I correct in thinking of it this way?

is it always short side over long side when making a proportion for similar triangles ? so if 10 was the radius of the small triangle the 10 would be on top ??
• yes, both ways you listed are correct

10/8=r/6, 8/10=6/r, and r/10=6/8 (what sal does in the video) are all valid ways of setting up this proportion. if you cross multiply & simplify, you should get the same answer (7.5) for any of these 3 ways.

there is no set rule for what to put on the top or bottom, as long as the whole thing is proportional. just pick whichever works best for you.
hope this helps :)
(1 vote)
• how do you calculate the volume of a tough figure
• Hello, is it possible when calculating the volume of grain inside the cone, to calculate the volume the cone holds in total and multiplying by 6/8 or 3/4 the portion of the cone filled up with grain? It's at that he announces the similar triangle method, I just wanted to know if the method I explained above worked as well.
(1 vote)
• You are partially correct. Similar volumes can be determined through ratio of corresponding edges. However, multiplying by (6 / 8) isn't correct.

Note the following formula (It can be helpful sometimes)

For 2D similar polygon area A_p and A_q with corresponding edges p and q.
A_p / A_q = (p / q)²

For 3D similar polyhedral volume V_p and V_q with corresponding edges p and q.
V_p / V_q = (p / q)³

Now, lets try using this formula!
Volume of cone = (1 / 3) * 10² * 8 * π = 837.8 m³
Let the volume of grain be V.
V / 837.8 = (6 / 8)³
V = 353.4 m³
(1 vote)