A spherical ball weighs three times as much as another ball of identical appearance and composition. The second ball weighs less because it is actually hollow inside. Find the radius of the hollow cavity in the second ball, given that each ball has a 5-inch radius.
since they have the same density, we have to worry about is the volume, since it is in the same ratio as the weight.
If the outside radius is R and the inside radius of the cavity is r, then you have
4/3 π * R^3 = 3 * 4/3 π(R^3 - r^3)
now solve for r.
Well, I have to say, these balls are quite deceptive! It sounds like the second ball is trying to pull off a magic trick!
Let's call the mass of the solid ball "M" and the mass of the hollow ball "m." Now, we know that the spherical ball weighs three times as much as the hollow ball, so we can write an equation:
M = 3m
We can also use the formula for the mass of a sphere to relate the masses to their radii:
M = (4/3)πr^3
m = (4/3)π(R^3 - r^3)
where "r" is the radius of the hollow part and "R" is the overall radius of the hollow ball, which is given as 5 inches.
Substituting these equations into our initial equation:
(4/3)π(5^3) = 3((4/3)π((5^3)-(r^3)))
Now, let's simplify:
500π = 3((500π)-(r^3))
Dividing through by 3π:
500 = 500 - r^3
And solving for r:
r^3 = 0
Now, this is where things get a little tricky. It turns out that the hollow ball with a 5-inch radius and a hollow cavity has no radius for the hollow part! It's empty inside, so the radius of the hollow cavity is zero.
Remember, friends, when you're dealing with clowns or hollow balls, things might get a little odd. Always expect the unexpected!
To solve this problem, we need to use the principle of density. The density of a material is defined as its mass divided by its volume. Since both balls have the same appearance and composition, their densities are equal.
Let's assume the mass of the solid ball (ball A) is M and the mass of the hollow ball (ball B) is also M. It is given that the mass of ball A is three times the mass of ball B.
We know that the volume of a sphere is (4/3)πr^3, where r is the radius. Since the two balls have radii of 5 inches, we can calculate their volumes.
Volume of ball A = (4/3)π(5)^3 = (500/3)π
Volume of ball B = (4/3)π(r_hollow)^3, where r_hollow is the radius of the hollow cavity
Since the densities of both balls are the same, we can set up the following equation:
Mass of ball A / Volume of ball A = Mass of ball B / Volume of ball B
M / [(500/3)π] = M / [(4/3)π(r_hollow)^3]
Simplifying the equation by canceling out common terms and cross multiplication, we get:
(500/3) = 4(r_hollow)^3
Dividing both sides by 4 and multiplying by 3/500, we can isolate (r_hollow)^3:
(r_hollow)^3 = (3/500)*(500/4) = 3/4
Taking the cube root of both sides, we find:
r_hollow = (3/4)^(1/3)
Approximately, r_hollow ≈ 0.897 inches
Therefore, the radius of the hollow cavity in the second ball is approximately 0.897 inches.
To solve this problem, we need to find the radius of the hollow cavity in the second ball.
Let's denote the radius of the hollow cavity as "r". Since both balls have a 5-inch radius, the radius of the second ball can be expressed as the difference between the outer radius and the inner radius, which is (5 - r).
We are given that the spherical ball weighs three times as much as the hollow ball.
To solve the problem, we can use the following formula for the mass of a sphere:
M = (4/3)πr^3ρ,
where M is the mass, r is the radius, and ρ is the density.
Since the two balls have identical appearance and composition, their densities are the same. Therefore, we can compare the masses of the two balls using the equation:
(4/3)π(5^3)ρ = 3 * (4/3)π[(5 - r)^3]ρ,
where (4/3)π(5^3)ρ is the mass of the first ball, and 3 * (4/3)π[(5 - r)^3]ρ is the mass of the second ball.
By simplifying the equation, we get:
(5^3) = 3 * [(5 - r)^3].
Now, we can solve for "r" by rearranging the equation:
125 = 3 * [(5 - r)^3].
Dividing both sides by 3, we have:
125/3 = (5 - r)^3.
Taking the cube root of both sides, we get:
(125/3)^(1/3) = 5 - r.
Now, by subtracting 5 from both sides, we can solve for "r":
r = 5 - (125/3)^(1/3).
Using a calculator, we can find the approximate value of r.