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Volume of 3D shapes: Comparing, rates of flow

Volume of 3D shapes: Comparing, rates of flow

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Explainer Video

Tutor: Bilal

Summary

Volume of 3D shapes: Comparing, rates of flow

In a nutshell

You can apply your knowledge of the volumes of 3D shapes to approach more complicated problems that involve comparing volumes and rates of flow.



Comparing volumes

You may be asked to calculate the volumes of two 3D shapes and compare them. To solve these types of questions, always use the volume formulae and compare after working out the volume for each shape.


Example 1

A hemisphere of radius RRR has the same volume as a sphere of radius rrr. What is the value of Rr\dfrac{R}{r}rR​?


First, find the volumes of the two shapes.

The volume of the sphere is given by the formula:

​Vsphere=43πr3V_{sphere}=\dfrac{4}{3}\pi r^3Vsphere​=34​πr3​​


A hemisphere is a half of a sphere. This means that the volume of the hemisphere is:

​Vhemisphere=12×43πR3=23πR3V_{hemisphere}=\dfrac{1}{2}\times\dfrac{4}{3}\pi R^3=\dfrac{2}{3}\pi R^3Vhemisphere​=21​×34​πR3=32​πR3​​


It is given that the two volumes are equal. So, set these two expressions equal to one another:

​Vsphere=VhemisphereV_{sphere}=V_{hemisphere}Vsphere​=Vhemisphere​​​


​43πr3=23πR3\dfrac{4}{3}\pi r^3=\dfrac{2}{3}\pi R^334​πr3=32​πR3​​


​4r3=2R34r^3=2R^34r3=2R3​​


​R3r3=2\dfrac{R^3}{r^3}=2r3R3​=2​​


​(Rr)3=2\bigg( \dfrac{R}{r} \bigg)^3=2(rR​)3=2​​


​Rr=23‾\underline{\dfrac{R}{r}=\sqrt[3]{2}}rR​=32​​​​



Rates of flow

Rates of flow questions involve a 3D shape that has liquid flowing in or out of the shape at a constant rate. To appoach these questions, use the formula:


​rate of flow=volumetime\text{rate of flow}=\dfrac{\text{volume}}{\text{time}}rate of flow=timevolume​​​


Example 2

A cubic tank fills with water at a rate of 15cm3/s15cm^3/s15cm3/s. It takes 303030 minutes for the tank to fill completely. What is the length of one side of the cube?


First, make sure all the units are consistent. Convert 303030 minutes to seconds:

​30×60=180030\times60=180030×60=1800​ seconds


Now, rearrange the formula for the volume, and substitute in the values:

​rate of flow=volumetime⇒volume=time×rate of flow\text{rate of flow}=\dfrac{\text{volume}}{\text{time}}\Rightarrow \text{volume}=\text{time}\times\text{rate of flow}rate of flow=timevolume​⇒volume=time×rate of flow​​


​V=1800×15=27000V=1800\times15=27000V=1800×15=27000​​


​V=27000cm3V=27000cm^3V=27000cm3​​


Use the formula for the volume of a cube to work out the length of one side:

​V=x3=27000V=x^3=27000V=x3=27000​​

​x=270003=30cmx=\sqrt[3]{27000}=30cmx=327000​=30cm​​


The length of one side is 30cm‾\underline{30cm}30cm​.


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Calculating the volume of cubes and cuboids

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Calculating the volume of cubes and cuboids

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Volume of 3D shapes: Comparing, rates of flow

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Volume of 3D shapes: Comparing, rates of flow

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FAQs - Frequently Asked Questions

How do you solve rate of flow problems?

First, make sure all the units are consistent with each other. Then, rearrange the rate of flow equation for the desired quantity and substitute in the known values.

What is the formula for rate of flow?

The formula for rate of flow is rate of flow = volume/time.

How do you solve problems about comparing volumes?

Work out the volume of each shape using their respective formula. Then, compare the volumes.

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