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# 3d geometry

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If the diameter of a right cylindrical can with circular bases is increased by $$36\%$$, by what percent should the height be decreased in order to preserve the volume of the original can?

Apr 16, 2022

#1
+124696
+2

Let the original volume =  pi * r^2 * original height

Note that if the diameter is increased by 36%, then so is the radius

So

pi * r^2 *  original height      =   pi *  (1.36r) ^2  * new height

original height =   (1.36)^2  * new height

original height / (1.36)^2   =new height

original height  * ( 1 /1.36^2)  =  new height

.54 original height = new height

The original height  should be reduced by  ≈   (1 - .54)  =  .46 ≈  46%

Apr 16, 2022
edited by CPhill  Apr 16, 2022
#2
+2532
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Without loss of generality, let the diameter be 100, and let the height be 100.

The original volume is: $$250000 \pi$$

With the increased base, the volume of the new cube is: $$462400\pi$$

For the volume to stay the same, we have to multiply the height by $${250000 \over 462400} = {625 \over 1156}$$

This means that the percent decrease in the height is $$\approx \color{brown}\boxed{45.93 \text{%}}$$

Apr 16, 2022
#4
+2532
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BuilderBoi  Apr 16, 2022
#5
+9461
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Nice!

MaxWong  Apr 16, 2022
#6
+124696
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Whoopee !!!

Also.....good to see you on here, Max  !!!!!

CPhill  Apr 16, 2022
#7
+9461
+1

Nice to see you too, CPhill :)

It has been a long time since I was last active on this site. It feels great to come back to this site and see that it is still a very active forum.

MaxWong  Apr 16, 2022
#8
+124696
+1

I was away about nine months, too....it was a little slow when I first came  back, but it appears to have "revived" some !!!!

CPhill  Apr 16, 2022
#3
+9461
+1

Let $$h$$ be the original height and suppose a decrease of $$k\%$$ is required.

Let $$r$$ be the original radius,

Now,

$$\pi r^2 h = \pi (r(1 + 36\%))^2(h(1 - k\%))\\ (1.36)^2 (1 - k\%) = 1\\ k\% = 1 - \dfrac1{1.36^2}\\ k\% = \dfrac{531}{1156}\\ k = \dfrac{13275}{289}$$

Therefore, the height needs to be decreased by $$\dfrac{13275}{289}\%$$, which is approximately $$45.9\%$$.

Apr 16, 2022