John is balancing solid silver spheres on a scale. The smaller sphere has one quarter of the radius of the larger sphere. The larger sphere is placed on the right side of the scale as seen in the diagram.
How many total smaller spheres must be placed on the left side of the scale to keep the scale balanced?
Use
π=3.14
.
Answer Choices
4
8
16
64
Basically, you need to have an equal mass on both sides. Since the material is the same and the density is constant, this means you need an equal volume on either side.
The volume of a sphere is given by:
(4/3)*π*(r^3), where r is its radius.
So, the volume is proportional to r^3
This means, if you have a radius four times as large as a small sphere, the volume will be 4^3 = 64 times larger.
So, you need 64 small spheres to balance the larger one.
To determine the number of smaller spheres needed to balance the scale, we need to consider the volume of each sphere.
The volume of a sphere can be calculated using the formula V = (4/3)πr^3, where V is the volume and r is the radius.
Given that the smaller sphere has a radius one quarter that of the larger sphere, we can say that the radius of the smaller sphere is r/4.
Since we want to find the number of smaller spheres needed to balance the larger sphere, we can equate the volumes of the larger sphere and the combined smaller spheres.
Let's assume that the larger sphere has a radius of r.
The volume of the larger sphere is then given by V_large = (4/3)πr^3.
The total volume of the smaller spheres is given by V_total = (4/3)π(r/4)^3 * n, where n is the number of smaller spheres.
To keep the scale balanced, the volume of the larger sphere must be equal to the total volume of the smaller spheres. So we can equate the two equations:
(4/3)πr^3 = (4/3)π(r/4)^3 * n
Canceling out common terms and simplifying, we get:
r^3 = (r/4)^3 * n
Taking the cube root of both sides, we have:
r = (r/4) * ∛n
Dividing both sides by r/4, we get:
4 = ∛n
Cubing both sides, we have:
64 = n
Therefore, we need a total of 64 smaller spheres on the left side of the scale to balance the larger sphere.
So the correct answer is 64.