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?
A) 32 fl oz B) 64 fl oz C) 128 fl oz D) 256 fl oz

the volume of two similar solids is proportional to the cubes of their corresponding sides

so x/16 = 2^3/1^3
x/16 = 8/1
....
...

What do you think?

First, we need to determine the dimensions of the large can. Since the large can has twice the base circumference of the small can, its circumference is 2 times greater.

Let's assume the circumference of the small can is C. Then, the circumference of the large can is 2C.

Now, let's find the radius of the small can. The formula for the circumference of a circle is C = 2πr, where C is the circumference and r is the radius.

Since the circumference of the small can is C, we can rearrange the formula to solve for the radius: r = C / (2π).

Now, let's find the radius of the large can. Since the large can has twice the circumference of the small can, its circumference is 2C. Using the same formula as above, the radius of the large can is r = 2C / (2π) = C / π.

Now that we have the radius of both cans, we can calculate the volumes. The formula for the volume of a cylinder is V = πr^2h, where V is the volume, r is the radius, and h is the height.

Since both cans have the same height (not mentioned in the question), we can focus on the comparison of the volumes based on the radii.

The volume of the small can is V_small = π(r_small)^2h, and the volume of the large can is V_large = π(r_large)^2h.

Substituting the radii into the volume formulas, we get:

V_small = π[(C / (2π))]^2h = (C^2 / (4π))h

V_large = π[(C / π)]^2h = (C^2 / π)h

Since the second can is similar to the first, their volumes should be in the same ratio. In other words:

V_large / V_small = (C^2 / π)h / (C^2 / (4π))h

The height (h) cancels out, leaving us with:

V_large / V_small = 4

Therefore, the volume of the large can (V_large) is 4 times the volume of the small can (V_small). Since the small can contains 16 fl oz, the large can contains:

V_large = 4 * V_small = 4 * 16 fl oz = 64 fl oz

Therefore, the juice in the large can is 64 fl oz, which corresponds to option B.

To solve this problem, we need to understand the relationship between the volumes of two similar right circular cylinders.

Let's denote:
V1 = volume of the small can of juice
V2 = volume of the large can of juice

The volume of a right circular cylinder can be calculated using the formula: V = πr^2h, where r is the radius of the base and h is the height.

Since the two cans are similar, their ratios of corresponding dimensions will be the same. We know that the large can has twice the base circumference of the small can.

The circumference of a circle is given by the formula: C = 2πr, where r is the radius.

Let's denote:
C1 = circumference of the small can base
C2 = circumference of the large can base

Since C2 is twice C1, we have the equation: C2 = 2C1.

Further, the relationship between the circumferences and the radii is:
C1 = 2πr1 and C2 = 2πr2, where r1 is the radius of the small can and r2 is the radius of the large can.

Now, let's substitute C1 and C2 in terms of r1 and r2 respectively into the equation C2 = 2C1:
2πr2 = 2(2πr1)
r2 = 2r1

Since the cans are right circular cylinders, their base areas are given by the formula A = πr^2.

Thus, the ratio of the base areas of the large can and the small can will be:
A2/A1 = (πr2^2)/(πr1^2) = r2^2/r1^2

Now, let's substitute r2 = 2r1 into the equation of the ratio of base areas:
A2/A1 = (2r1)^2/r1^2 = 4(r1^2/r1^2) = 4

This tells us that the base area of the large can is 4 times that of the small can.

Finally, we can conclude that the volume of the large can V2 is 4 times the volume of the small can V1.

Since the small can contains 16 fl oz, the large can will contain: 4 x 16 fl oz = 64 fl oz.

Therefore, the answer is B) 64 fl oz.