Okay...here's a hard math problem!!!?

Question

A hollow metal ball has a wall that is 3 cm thick. If the total volume of the metal used is 684п (< п is supposed to be a symbol for pi) cubed centimeters, what is the radius of the ball?
I am very confused.

Answer 1

The volume of the metal is simply the volume of the sphere encompassing the outside of the ball minus the volume of the hollow region inside the ball. The radius of the hollow region is 3 cm less than the radius of the ball itself (the wall is 3 cm thick) If the radius of the ball is r, then this means 684π=4/3πr³-4/3π(r-3)³. Solving for r:

684π=4/3πr³-4/3π(r-3)³

Divide by 4/3π:

513=r³-(r-3)³

Expand (r-3)³

513=r³-(r³-9r²+27r-27)

Simplify:

513=9r²-27r+27

Divide both sides by 9:

57=r²-3r+3

Subtract 3 from both sides:

54=r²-3r

Add 9/4 to both sides:

225/4=r²-3r+9/4

Factor:

225/4=(r-3/2)²

Take the square root:

r-3/2=±15/2

Add 3/2:

r=3/2±15/2

Thus r=9 or r=-6. However, a negative radius is unphysical, therefore the correct solution is:

r=9 cm

Answer 2

The equation for the volume of a sphere is
v= 4/3 Pi r^3

So what we have is a hollow sphere, so the volume would be

v=(4/3 Pi ro^3) - (4/3 Pi ri^3)

where ro is the outer radius, and ri the inner radius; which is to say we find the volume of the sphere as if it was solid, and remove the hollow part.

We know that ro = ri + 3, and that v is 684*Pi.

So, let's work out the equation a little.

v=(4 Pi ro^3) - (4 Pi ri^3)
v = 4/3 Pi (ro^3 - ri^3)

and we can replace ro with (ri+3)
v= 4/3 Pi ((ri+3)^3 - ri^3)

expanding (ri+3)^3, wich is equal to (ri^3 + 9 ri^2 + 27 ri + 27), we then have

v =4/3 Pi (9 ri^2 + 27 ri + 27) = 684 Pi

513 = 9 ri^2 + 27 ri + 27
57 = ri^2 + 3 ri + 3
0 = ri^2 + 3 ri - 54

which is a quadratic equation with two roots: one at -9 and one at 6 (the negative one makes no sense in the current context, so we retain 6 as the real inner radius).

The outer radius is thus 6 + 3, or 9 cm.

Answer 3

OK, first lets define some terms:
V= Total volume
V1= Volume inside metal ball

Formula:
V=(4/3)*n*r^3 (n=pi)

Known terms:
V=684n
Wall-thickness: 3 cm

684n = (4/3)*n*r^3 I Multiply across with 3

2052n = 4*n*r^3 I Divide across by 4*n

513 = r^3 I Take the cubic root across [a^/(1/3)]

r = 8 cm

The outer radius of the metal ball is 8 cm.

In order to find the radius of the inside of the ball:

Subtract 8 - 3 (Outside radius - thickness of wall) = 5 cm

Place appropriate numbers into equation in order to obtain the volume inside the ball:

V = (4/3)*n*5^3

V = 167n (166.66666666n)

Answer 4

The volume of the metal is the difference between the outer volume and the inner volume of the sphere,
Hence V = 4/3pi((R^3 -(R-3)^3)
684pi = 4/3pi((R^3 -(R^3 -9R^2 + 27R -27))
684pi = 4/3pi((R^3 -R^3 + 9R^2 - 27R + 27)
684pi = 4/3pi(9R^2 - 27R +27)
cancelling pi and crossmultiplying left hand and right terms
684x3/4 = 9R^2 -27R + 27
513 = 9(R^2 -3R + 3)
or
513/9 = R^2 -3R + 3
57 = R^2 -3R + 3
or
R^2 - 3R -54 = 0
Factorizing
R^2 - 9R + 6R -54 = 0
which gives R(R - 9) + 6( R - () = 0
or (R - 9)(R + 6) = 0
So the solutions for the radii are R = 9cm or R = -6cm
The negative answer is clearly inadmissible
Hence R = 9cm

Answer 5

Try not to get lost.
I think it's 7.5
because the volume is the amount of metal used
so 684 times pi(short version of pi equals 3.14) equals 2147.76
Then, because the metal ball has 3cm of metal
divide (684x3.14) by 3 which equals 715.92
That would be at least about the surface area.
The equation for surface area is
Surface area = 4 times pi times the radius squared
So:
715.92 = (4) (3.14) (radius squared)
715.92 = 12.56 (radius squared)
divide 12.56 from both sides
57 = radius squared
Then, "unfactor" it.
I got 7.549834435.
Then, I would round to the nearest tenth,which is 7.5

Answer 6

The volume of any sphere is (4.3) (pi)r^3

(4/3) (pi)r^3 = 684(pi)

So....solve for "r". What's the big deal?

r^3 = 684(pi)/(4/3)(pi)

==> r^3 = 3*684/4

==> r = Cubed root[(3*684)/4]

Can you grab a calculator and do the rest from here.

Answer 7

use the equation for the volume of a sphere, plug in what you know, and then take your answer and subtract another sphere volume with a diameter that's 6cm less than your first ( which compensates for the fact that it's hollow->its wall is 3cm thick on both sides)

Answer 8

For a sphere V=(4/3)pi*r^3
the volume here is the difference between the 2 spheres.
r2-r1=3cm
V=((4/3)pi(r2^3-r1^3)
r2=r1+3cm
r2^3=(r1+3cm)(r1^2+6r1+9)=r1^3+6r1^2+9r1
3r1^2+18r1+27)
r2^3=r1^3+6r1^2+3r1^2+9r1+18r1+27
r2^3=r1^3+9r1^2+27r1+27=r1^3+9(r1^2+3r1+3)
r2^3-r1^3=9(r1^2+3r1+3)
V=684pi=(4/3)pi(9)(r1^2+3r1+3)
76=(4/3)(r1^2+3r1+3)

57=r1^2+3r1+3
r1^2+3r1-54=0
r1=(-3+sqrt(9+4*54))/2
r1=(-3+sqrt(225))/2=(-3+15)/2=15/2=6
r2=6+3=9cm
so the radius is 9cm.

check

the volume of metal is
V=(4/3)pi(r2^3-r1^3)=(4/3)pi(9^3-6^3)=(4/3)pi(729-216)
V=(4/3)pi*513=684pi

Answer 9

the volume of a sphere is V = 4/3 * pi * r^3

so, 684 = 4/3 * pi * r^3
513 = pi * r^3
r^3 = 163.2929
r = 163.2929^(1/3) = 5.4658

Answer 10

well it would be 3cm x 684n, and pi = 2.34 so you would do 684 x 2.34 and get 1600.56 and divide it by 3, getting 533.52, at least i think anyway...