Volume decreasing what is raidus
- Thread starter blink
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Are you saying you can do this: "If the radius increases by 20%, by how much does the volume increase?"
But you cannot do this: "If the volume decreases by 39%, but how much did the radius decrease?"
Okay, how do you do the 20% increase in the radius? It will help us on our pathway.
But you cannot do this: "If the volume decreases by 39%, but how much did the radius decrease?"
Okay, how do you do the 20% increase in the radius? It will help us on our pathway.
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Very good.
Notice, how you do not need to calculate it directly
We have \(\displaystyle \frac{4}{3}\cdot\pi r^{3} = V_{0}\)
What percent increase in volume is: \(\displaystyle \frac{4}{3}\cdot\pi (1.2r)^{3} = V_{1}\)
With just a little manipulation, \(\displaystyle \frac{4}{3}\cdot\pi\cdot 1.2^{3}r^{3} = 1.2^{3}\cdot V_{0} = V_{1}\)
And finally, \(\displaystyle 1.2^{3} = 1.728\) and we have the same correct result that you obtained.
Can you apply this methodology to the other problem type?
We have \(\displaystyle \frac{4}{3}\cdot\pi r_{0}^{3} = V\)
What percent decrease in radius is: \(\displaystyle \frac{4}{3}\cdot\pi r_{1}^{3} = (1 - 0.30)\cdot V = 0.70\cdot V\)
Notice, how you do not need to calculate it directly
We have \(\displaystyle \frac{4}{3}\cdot\pi r^{3} = V_{0}\)
What percent increase in volume is: \(\displaystyle \frac{4}{3}\cdot\pi (1.2r)^{3} = V_{1}\)
With just a little manipulation, \(\displaystyle \frac{4}{3}\cdot\pi\cdot 1.2^{3}r^{3} = 1.2^{3}\cdot V_{0} = V_{1}\)
And finally, \(\displaystyle 1.2^{3} = 1.728\) and we have the same correct result that you obtained.
Can you apply this methodology to the other problem type?
We have \(\displaystyle \frac{4}{3}\cdot\pi r_{0}^{3} = V\)
What percent decrease in radius is: \(\displaystyle \frac{4}{3}\cdot\pi r_{1}^{3} = (1 - 0.30)\cdot V = 0.70\cdot V\)
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What if we rephrased:
The Volume of a Sphere is Directly Proportional to the Cube of the Radius. The constant of proportionality is (4/3)pi.
\(\displaystyle \frac{0.70\cdot V}{V} = \frac{\frac{4}{3}\cdot \pi r_{1}^{3}}{\frac{4}{3}\cdot \pi r_{0}^{3}} = \frac{r_{1}^{3}}{r_{0}^{3}} = 0.70\)
Then \(\displaystyle \frac{r_{1}}{r_{0}} = \sqrt[3]{0.70} = 0.8879\)
Don't try to cram it all in your head. Let the notation help you.
The Volume of a Sphere is Directly Proportional to the Cube of the Radius. The constant of proportionality is (4/3)pi.
\(\displaystyle \frac{0.70\cdot V}{V} = \frac{\frac{4}{3}\cdot \pi r_{1}^{3}}{\frac{4}{3}\cdot \pi r_{0}^{3}} = \frac{r_{1}^{3}}{r_{0}^{3}} = 0.70\)
Then \(\displaystyle \frac{r_{1}}{r_{0}} = \sqrt[3]{0.70} = 0.8879\)
Don't try to cram it all in your head. Let the notation help you.
After some time i hope i have deeper understanding for it also. thx alot for your time!