can someone please help me solve this problem with the steps.
A small can of juice contains 16 fl oz. A large can has twice the base circumference of the first. How much juice is in the large can if both cans are right circular cylinders and the second can is similar to the first?
To find the amount of juice in the large can, we need to first determine the dimensions of the large can based on the given information about the small can.
Step 1: Understand the problem.
We are given that the small can of juice contains 16 fl oz and that the large can has twice the base circumference of the first. We also know that both cans are right circular cylinders and that the second can is similar to the first.
Step 2: Define the variables.
Let's assign variables to the dimensions of the small can:
- r1: the radius of the base of the small can
- h1: the height of the small can
We also need to define the corresponding variables for the large can:
- r2: the radius of the base of the large can
- h2: the height of the large can
Step 3: Find the relationship between r1 and r2.
Since the large can is similar to the small can, their corresponding dimensions will be proportional (keeping the same shape). Therefore, we can say that the ratio of the radii of the small and large cans is equal to the ratio of their circumference.
The circumference of a circle is given by the formula: C = 2πr
For the small can:
C1 = 2πr1
For the large can:
C2 = 2πr2
Given that the large can has twice the base circumference of the small can:
C2 = 2 * C1
Replacing the circumferences with their formulas:
2πr2 = 2 * (2πr1)
r2 = 2 * r1
So, we have found that the radius of the large can is twice the radius of the small can.
Step 4: Find the relationship between h1 and h2.
Since the large can is similar to the small can, their corresponding dimensions will be proportional. Therefore, we can say that the ratio of the heights of the small and large cans is equal to the ratio of their circumferences.
For the small can, the circumference is equal to the height:
C1 = 2πr1 = h1
For the large can, the circumference is given by 2πr2, so the height of the large can is also 2πr2.
Therefore, h2 = 2πr2
Step 5: Calculate the amount of juice in the large can.
The amount of juice in a can is directly proportional to the volume of the can. The volume of a right circular cylinder is given by the formula: V = πr^2h
For the small can:
V1 = πr1^2h1
For the large can, we need to use the relationships we found earlier:
V2 = πr2^2h2
Substituting the relationships:
V2 = π(2r1)^2(2πr2)
Simplifying:
V2 = 4πr1^2 * (2πr2)
V2 = 8π^2r1^2r2
Step 6: Calculate the radius of the small can.
We were not given the radius of the small can, so we need to find it. We can use the volume of the small can to find the radius.
Given that the small can contains 16 fl oz, we know that its volume is 16 fl oz or 16 cubic inches.
V1 = 16π
Substituting the values into the volume formula:
16π = πr1^2h1
Since we were not given the height of the small can, we cannot solve for r1.
Therefore, without the information about the height of the small can, we cannot determine the amount of juice in the large can based on the given information.