Mass, Volume, Density work with a Sphere

In summary, if you increase a, the mass of the sphere decreases, but if you decrease a, the mass of the sphere increases. The maximum mass of the sphere is found when a is reduced to 0.
  • #1
jdsconsumer
7
0
A spherical shell has an outside radius of 2.75 cm and an inside radius of a. The shell wall has uniform thickness and is made of a material with density 4.59 g/cm3. The space inside the shell is filled with a liquid having a density of 1.00 g/cm3. (a) Find the mass m of the sphere, including its contents, as a function of a.

Answer that I got much help on from previous helpers: m = 399.33 - 4.79 pi ( a^3 ) grams

(b) For what value of a does m have its maximum possible value?

(c) What is this maximum mass?

(d) Explain whether the value from part (c) agrees with the result of a direct calculation of the mass of a solid sphere of uniform density made of the same material as the shell.

I am now on part (b). So am I trying to find the maximum of that function of "a"?
 
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  • #2
Is there any other
equation between a and m If you can tell me then I can help you
 
  • #3
jdsconsumer said:
find the maximum of that function of "a"
According to the equation, if you increase a what happens to m?
 
  • #4
haruspex said:
According to the equation, if you increase a what happens to m?

uhm, if a increases it looks like m will get smaller, right?
 
  • #5
ArmanCham said:
Is there any other
equation between a and m If you can tell me then I can help you

No other equation that is given to me. I think we have to formulate our own...
 
  • #6
jdsconsumer said:
uhm, if a increases it looks like m will get smaller, right?
Right. And if you decrease a?
 
  • #7
jbriggs444 said:
Right. And if you decrease a?

Then m would get larger.
 
  • #8
So if you want to maximize m, what must you do with a?
 
  • #9
jbriggs444 said:
So if you want to maximize m, what must you do with a?

make it smaller and smaller until I can't make it any smaller? So if I reduced it to 0, the minimum it's radius could be, then m couldn't get any larger at that point. At least from the decrement of a.
 
  • #10
Yup. There you go.
 
  • #11
jbriggs444 said:
Yup. There you go.

You are awesome, I love how you did that for me! Incredible!
Thank you, jbriggs444!
 
  • #12
grrr, why was that so hard for me to talk through on my own?! That is awesome! I feel so much better now!
 

What is the formula for calculating the mass of a sphere?

The formula for calculating the mass of a sphere is mass = density x volume. The density of the sphere can be found by dividing the mass by the volume.

How do you measure the volume of a sphere?

The volume of a sphere can be measured by using the formula V = (4/3)πr³, where r is the radius of the sphere. Alternatively, the volume can also be measured by submerging the sphere in water and measuring the displacement.

What is the relationship between mass, volume, and density of a sphere?

The mass, volume, and density of a sphere are all interrelated. The mass of a sphere is directly proportional to its volume, meaning that as the volume increases, so does the mass. Density, on the other hand, is inversely proportional to volume, so as the volume increases, the density decreases.

How does the density of a sphere affect its buoyancy?

The density of a sphere affects its buoyancy by determining whether it will float or sink in a fluid. If the density of the sphere is greater than the density of the fluid, it will sink. If the density of the sphere is less than the density of the fluid, it will float.

Can the mass and volume of a sphere be changed without changing its density?

No, the density of a sphere is a fixed value and cannot be changed without altering either the mass or volume. If the mass increases, but the volume remains the same, the density will increase. Similarly, if the volume increases, but the mass remains the same, the density will decrease.

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