Math Problem Statement
Find the radius of this cone.
What dimensions are given on this cone? What dimension are you trying to find?
r = ?
Area Label Arrowh = 12
volume = 16π
V =
3
1
πr
2
h
The volume and height are given. You're trying to find the radius . Good work! Start by substituting the volume and height in the equation. r = ? Area Label Arrowh = 12 volume = 16π V
3 1 π r 2 h
16π
3
1
· π · r
2
·
12
Great job!
We can use the commutative property to multiply the factors in any order. It could be helpful to multiply
3
1
and 12 first so that we can rewrite this expression with fewer terms. Rearrange the factors to show that we’re multiplying
3
1
and 12 first, then multiplying by π times r
2
.
r = ?
Area Label Arrowh = 12
volume = 16π
16π
3 1
· 12 · π · r 2
Great job!
Multiply
3
1
by 12 to rewrite the right side of the equation.
r = ?
Area Label Arrowh = 12
volume = 16π
V
3 1 πr 2 h
16π
3 1 · π · r 2 · 12
16π
3 1 · 12 · π · r 2
16π
· π · r 2
Good work! What is another way you can write 4 · π in the equation? r = ? Area Label Arrowh = 12 volume = 16π V
3 1 πr 2 h
16π
3 1 · π · r 2 · 12
16π
3 1 · 12 · π · r 2
16π
4 · π · r 2
16π
4π · r 2
You got it! Next, what move can we make to get rid of 4π so that r 2 is alone on the right? r = ? Area Label Arrowh = 12 volume = 16π V
3 1 πr 2 h
16π
3 1 · π · r 2 · 12
16π
3 1 · 12 · π · r 2
16π
4 · π · r 2
16π
4π · r 2
Divide both sides of the equation by 4π
Multiply both sides of the equation by 16
Divide both sides of the equation by r Excellent! You can divide both sides by 4π to get rid of 4π and get r 2 alone on the right side of the equation. Record dividing both sides by 4π in your equation work. r = ? Area Label Arrowh = 12 volume = 16π V
3 1 πr 2 h
16π
3 1 · π · r 2 · 12
16π
3 1 · 12 · π · r 2
16π
4 · π · r 2
16π
4π · r 2
4π
4π Good work! Now, divide both sides by 4π to get r 2 alone on the right side of the equation. r = ? Area Label Arrowh = 12 volume = 16π V
3 1 πr 2 h
16π
3 1 · π · r 2 · 12
16π
3 1 · 12 · π · r 2
16π
4 · π · r 2
16π
4π · r 2
4π 4π
= Great job! You found the value of r 2 is 4. What is the value of r? r = ? Area Label Arrowh = 12 volume = 16π V
3 1 πr 2 · h
16π
3 1 · π · r 2 · 12
16π
3 1 · 12 · π · r 2
16π
4 · π · r 2
16π
4π · r 2
4π 4π
4
r 2
= r
Solution
You're looking to find the radius of a cone, and you are given the following:
- The height
- The volume
Step-by-Step Solution:
The formula for the volume of a cone is:
Substitute the known values for and :
Next, simplify by multiplying and :
Now, divide both sides of the equation by to isolate :
Finally, take the square root of both sides to solve for :
Conclusion:
The radius of the cone is 2.
Would you like more details on any part of the solution or have other questions? Here are some follow-up questions you might find useful:
- How does the formula for the volume of a cone relate to the geometry of the cone?
- What happens to the radius if the volume or height changes?
- How do we derive the formula for the volume of a cone from the formula for the volume of a cylinder?
- What is the surface area of a cone and how can we calculate it?
- How do the dimensions of a cone impact its volume?
Tip: When solving for variables in formulas involving pi, make sure to simplify the pi terms early to make the calculations easier.
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Math Problem Analysis
Mathematical Concepts
Geometry
Volume of a Cone
Algebra
Formulas
V = (1/3)πr^2h
Theorems
-
Suitable Grade Level
Grades 7-9