A cylindrical tank with diameter 20in. is half filled with water. How much will the water level rise if a metallic ball of radius 4 in. is placed in the tank?

Since the tank has a cross-section area of 100π in^2,

each in^3 of ball raises the water level by 1/(100π) in.

So, figure the volume of the ball and multiply.

To find out how much the water level will rise when a metallic ball is placed in the tank, we can use the principle of displacement.

The volume of water displaced by the metallic ball is equal to the volume of the ball itself.

The volume of a sphere can be calculated using the formula:

V = (4/3)πr³

where V is the volume and r is the radius of the sphere.

In this case, the radius of the ball is given as 4 inches. Substituting this into the formula, we get:

V = (4/3)π(4)³
V = (4/3)π(64)
V = (4/3)(3.14159)(64)
V = 268.0825737 cubic inches (approximately)

Since the tank is initially half-filled with water, the initial volume of water is half the volume of the tank.

The volume of a cylinder can be calculated using the formula:

V = πr²h

where V is the volume, r is the radius of the base of the cylinder, and h is the height or length of the cylinder.

In this case, the diameter of the tank is given as 20 inches, so the radius of the tank is half of that, which is 10 inches.

The initial volume of water in the tank is:

V_initial = (1/2)(π(10)²h)
V_initial = (1/2)(3.14159)(100)h
V_initial = 157.0796327h cubic inches (approximately)

Now, let's assume the water level rises by a height h_rise when the metallic ball is placed in the tank.

The new volume of water in the tank is equal to the initial volume of water plus the volume of the metallic ball, so we have:

V_new = V_initial + V_ball
V_new = 157.0796327h + 268.0825737

We want to find the height h_rise, so we set the new volume of water equal to the volume of the tank, which is the volume of a cylinder:

V_new = π(10)²(h + h_rise)
157.0796327h + 268.0825737 = 3.14159(10)²(h + h_rise)
157.0796327h + 268.0825737 = 314.159h + 314.159h_rise

Now we can solve for h_rise:

314.159h_rise = 268.0825737 - 157.0796327h
h_rise = (268.0825737 - 157.0796327h) / 314.159

By substituting the initial height h of the water, you can calculate the value of h_rise.