Radius of 2nd Sphere: Find Mass 5x Greater than 1st

[afg_91320]

Homework Statement

Two spheres are cut from a certain uniform rock. One has a radius 4.50cm. The mass of the
other is 5 times greater. Find its radius

Homework Equations

r₁ = 4.50cm
r₂ = ? when mass is 5x greater

The Attempt at a Solution

Looking at this problem I first thought of somehow integrating the volume of a sphere
which is 4/3$$\pi$$r³. But then i though i was just thinking too hard. it made more sense for the radius to be 5x the radius as i figure the mass was proportional to the radius.

so: r₂ = 5(4.50cm) = 22.5cm

however i got this marked wrong as the answer was 7.69 cm. what wasn't I looking at clearly?

 

[berkeman]

Homework Statement

Two spheres are cut from a certain uniform rock. One has a radius 4.50cm. The mass of the
other is 5 times greater. Find its radius

Homework Equations

r₁ = 4.50cm
r₂ = ? when mass is 5x greater

The Attempt at a Solution

Looking at this problem I first thought of somehow integrating the volume of a sphere
which is 4/3$$\pi$$r³. But then i though i was just thinking too hard. it made more sense for the radius to be 5x the radius as i figure the mass was proportional to the radius.

so: r₂ = 5(4.50cm) = 22.5cm

however i got this marked wrong as the answer was 7.69 cm. what wasn't I looking at clearly?

You started on the right track. Mass is proportional to volume, so the ratio 5 applies to the two radii how?

 

[afg_91320]

You started on the right track. Mass is proportional to volume, so the ratio 5 applies to the two radii how?

well that would be proportional to the mass right? so if r2 is 5x in mass then r1 is 1/5(mass)? set up my equation to isolate r to get the radius...

 

[berkeman]

well that would be proportional to the mass right? so if r2 is 5x in mass then r1 is 1/5(mass)? set up my equation to isolate r to get the radius...

Mass is proportional to volume. Write the fraction V1/V2 out fully, and that will show you a ratio involving some form of the radii. That's where the 5x comes into play...

 

[rwisz]

It seems like you were on the right track with your initial approach of using the formula for the volume of a sphere. However, instead of looking at the relationship between the radii, you should have considered the relationship between the volumes and the masses. Since the mass of the second sphere is 5 times greater than the first, we can set up the following equation:

(4/3)pi(r1)^3 = 5(4/3)pi(r2)^3

Simplifying this equation, we get:

r2 = (r1)^3 * (5/4)^(1/3)

Plugging in the given value for r1 = 4.50cm, we get:

r2 = (4.50cm)^3 * (5/4)^(1/3) = 7.69cm

Therefore, the radius of the second sphere is 7.69cm. It is important to consider the relationship between the volumes and masses in order to find the correct solution.