The ratio of the volume of Cylinder A to the volume of Cylinder B is 1:5. Cylinder A is similar to Cylinder C with a scale factor of 1:2 and Cylinder B is similar to Cylinder D with a scale factor of 1:3. What is the ratio of the volume of Cylinder C to the volume of Cylinder D?
Solve and proof, following problem solving rubric.
based on :
ratio of volumes is proportional to the cube of their corresponding sides.
so ratio of volumes:
A : B = 1 : 5
A : C = 1^3 : 2^3 = 1 : 8
B : D = 1^3 : 3^3 = 1 : 27
A/B = 1/5
B = 5A
A/C = 1/8
C = 8A
B/D = 1/27
D = 27B
ratio of volume of C :D
= 8A : 27B
= (8/27)(A/B)
= (8/27)((1/5)
= 8 : 135
check my arithmetic
To solve this problem, we'll follow the given information and use the concept of similar figures and the property of volume ratios.
First, let's define some variables:
- Let V(A) represent the volume of Cylinder A.
- Let V(B) represent the volume of Cylinder B.
- Let V(C) represent the volume of Cylinder C.
- Let V(D) represent the volume of Cylinder D.
- Let k₁ represent the scale factor between Cylinder A and Cylinder C (1:2).
- Let k₂ represent the scale factor between Cylinder B and Cylinder D (1:3).
Now, let's use the given information to form equations:
1. Ratio of the volume of A to B is 1:5:
V(A) : V(B) = 1 : 5 [Equation 1]
2. Scale factor between Cylinder A and Cylinder C is 1:2:
V(A) = k₁^3 * V(C) [Equation 2]
3. Scale factor between Cylinder B and Cylinder D is 1:3:
V(B) = k₂^3 * V(D) [Equation 3]
To find the ratio of the volume of Cylinder C to Cylinder D (V(C) : V(D)), we need to eliminate V(A) and V(B) from the equations and rearrange them to isolate V(C) and V(D).
From Equation 1, we can rewrite V(B) in terms of V(A):
V(B) = 5V(A)
Substituting V(A) in Equation 2:
5V(A) = k₁^3 * V(C)
Dividing both sides by 5:
V(A) = (k₁^3 / 5) * V(C)
Substituting this expression for V(A) in Equation 3 to eliminate V(A):
(k₁^3 / 5) * V(C) = k₂^3 * V(D)
Now, rearrange the equation to isolate V(C) and V(D):
V(C) = (5 / k₁^3) * (k₂^3) * V(D)
So, the ratio of the volume of Cylinder C to Cylinder D is:
V(C) : V(D) = (5 / k₁^3) * (k₂^3) : 1
Since k₁ represents the scale factor between Cylinder A and Cylinder C (1:2) and k₂ represents the scale factor between Cylinder B and Cylinder D (1:3), we can substitute their values into the ratio expression.
Finally, the ratio of the volume of Cylinder C to Cylinder D is:
V(C) : V(D) = (5 / (1^3)) * ((1^2)^3) : 1
= 5 * 1^3 : 1^1
= 5 : 1
Therefore, the ratio of the volume of Cylinder C to Cylinder D is 5:1.
To find the ratio of the volume of Cylinder C to the volume of Cylinder D, we need to first find the individual volumes of each cylinder.
Let's assign variables to the unknown values:
- Let x be the ratio of the volumes between Cylinder A and Cylinder C.
- Let y be the ratio of the volumes between Cylinder B and Cylinder D.
Given information:
- The ratio between the volume of Cylinder A and Cylinder B is 1:5.
- The ratio between the scale factor of Cylinder A and Cylinder C is 1:2.
- The ratio between the scale factor of Cylinder B and Cylinder D is 1:3.
Step 1: Determine the ratio between Cylinder A and Cylinder C
Since their scale factors have a ratio of 1:2, the ratio of their volumes will also be 1:2.
Therefore, x = 1:2
Step 2: Determine the ratio between Cylinder B and Cylinder D
Since their scale factors have a ratio of 1:3, the ratio of their volumes will also be 1:3.
Therefore, y = 1:3
Step 3: Determine the ratio between Cylinder C and Cylinder D
To find the ratio between Cylinder C and Cylinder D, we need to use the information from the ratios of Cylinder A to Cylinder C and Cylinder B to Cylinder D.
The ratio of the volume of Cylinder A to Cylinder B is 1:5, and the ratio of Cylinder A to Cylinder C is 1:2.
This means that the ratio of Cylinder B to Cylinder D is also 1:5, based on their similar ratios.
Since Cylinder C and Cylinder D have the same ratios as Cylinder A and Cylinder B, it means that the ratio between Cylinder C and Cylinder D is the same as the ratio between Cylinder A and Cylinder B, which is 1:5.
Therefore, the ratio of the volume of Cylinder C to the volume of Cylinder D is 1:5.
Proof:
Let's use an example to verify the solution.
Assume the volume of Cylinder A is 1 unit and the volume of Cylinder B is 5 units, based on the given ratio.
Since the scale factor between Cylinder A and Cylinder C is 1:2, the volume of Cylinder C would be 2 units.
Similarly, since the scale factor between Cylinder B and Cylinder D is 1:3, the volume of Cylinder D would be 15 units.
The ratio of the volume of Cylinder C to the volume of Cylinder D is 2:15, which can also be simplified to 1:5.
Therefore, the solution is verified and the ratio of the volume of Cylinder C to the volume of Cylinder D is indeed 1:5.