Two spheres are used in a certain machine. One has a volume of 7 cm3 and the other has a volume of 189 cm3. The radius of the small sphere is what fraction of the radius of the large sphere? Which of the following represents the first step to solve this problem?

Question options:

189 ÷ 7
73
189√3
189 ÷ 3

I think option 4 but it says that I'm incorrect. Please help.

the volumes of the two spheres would be proportional to the cube of their radii.

so
r1^3 / r2^3 = 7 / 189 , where r1 is smaller and r2 is the larger radius
(r1/r2)^3 = 7/189 = 1/27
r1 / r2 = 1/3

the radius of the smaller is 1/3 of the radius of the larger.
None of the given choices are correct.

What did you do first?

Did you not look at my solution?

you could do it the long way .....

(4/3)π r1^3 = 189
r2^3 = 189(3/4)(1/π) = 567/4π
r2 = (567/4π)^(1/3) = 3.56006...

similarly
(4/3)π r2^3 = 7
r1 = ...
= 1.186687...

r2/r1 = 3.560063... / 1.186687.. = 3/1
so the smaller radius is 1/3 the larger

Ok, thank you :). Sorry for the trouble.

To solve this problem, we need to find the ratio between the radii of the two spheres. The ratio of the volumes of two spheres is equal to the cube of the ratio of their radii.

Let's represent the radius of the small sphere as "r" and the radius of the large sphere as "R". The volume of a sphere can be calculated using the formula: V = (4/3)πr^3.

Given that the volume of the small sphere is 7 cm^3, we have:

(4/3)πr^3 = 7

Similarly, the volume of the large sphere is 189 cm^3, so we have:

(4/3)πR^3 = 189

To find the ratio of the radii, we can divide the second equation by the first equation:

[(4/3)πR^3] / [(4/3)πr^3] = 189 / 7

Simplifying and canceling like terms, we get:

(R^3 / r^3) = 189 / 7

Taking the cube root of both sides, we have:

(R / r) = (189 / 7)^(1/3)

Now, let's look at the options provided:

1. 189 ÷ 7: This option does not represent the correct equation we derived above. It would give us the ratio of the volumes and not the ratio of the radii.

2. 73: This value is not related to the problem statement and does not represent any step in solving the problem.

3. 189√3: This option does not represent the correct equation we derived above. It does not provide a valid relationship between the radii of the two spheres.

4. 189 ÷ 3: This option represents the equation (189 / 7)^(1/3) correctly. It is the correct first step to solve this problem.

Therefore, the correct first step to solve this problem is option 4, not option 1.