A heavy 9cm diameter ball is placed in an empty cylindrical tin
of diameter 12cm. Enough water is poured into the can to cover
the ball.if the ball is then removed , how far does the water
level fall
volume of ball = (4/3)π(4.5)^3
So that is the change in volume in the cylinder once removed
π(6^2) h = (4/3)π(9/2)^3
36h = (4/3)(729/8) = 243/2
h = 243/72 = 27/8
the water will fall 27/8 cm or 3.375 cm
check my arithmetic.
Well, it really depends on how thirsty the water is. It might fall a little for a casual sip or a lot if it's feeling particularly parched. I guess we can calculate the exact amount using a bit of math!
First, let's calculate the volume of water in the can with the ball inside. The volume of a cylinder can be found using the formula V = πr^2h, where r is the radius and h is the height. The radius of the can is half of its diameter, so it's 6 cm. Since the ball is 9 cm in diameter, its radius is 4.5 cm.
The height of the water in the can is the same as the height of the can itself, so we need to find the height. The height can be calculated by subtracting the ball's radius from the can's radius.
h = 6 cm - 4.5 cm
h = 1.5 cm
Now, we can find the volume of the water in the can with the ball inside:
V_1 = π(6 cm)^2(1.5 cm)
V_1 = 169.65 cm^3
Next, let's calculate the volume of just the ball:
V_2 = (4/3)π(4.5 cm)^3
V_2 = 381.7 cm^3
Finally, we subtract the volume of the ball from the initial volume of the water to find how much the water level falls:
V_fall = V_1 - V_2
V_fall = 169.65 cm^3 - 381.7 cm^3
V_fall = -212.05 cm^3
Uh oh! It seems the volume of water actually increases when the ball is removed. That's because the ball displaces some water, and when it's removed, the water rushes back to fill that space. So, the water level doesn't fall at all. It's a bit like a surprise party for the ball!
To determine how far the water level falls when the ball is removed, you need to calculate the volume of the ball and compare it to the volume of water in the can.
1. Calculate the volume of the ball:
The volume of a sphere can be calculated using the formula: V = (4/3)πr^3,
where V is volume and r is the radius of the sphere.
The diameter of the ball is given as 9 cm, so the radius (r) is half of the diameter, which is 9/2 = 4.5 cm.
Substituting the radius into the formula, the volume of the ball is:
V = (4/3)π(4.5^3) = 381.7 cm^3 (rounded to one decimal place).
2. Calculate the volume of water in the can:
The volume of a cylinder can be calculated using the formula: V = πr^2h,
where V is volume, r is the radius of the cylinder, and h is the height of the cylinder.
The diameter of the can is given as 12 cm, so the radius (r) is half of the diameter, which is 12/2 = 6 cm.
To find the height of the water, we need to consider that when the ball is placed in the tin, it displaces some water (equal to the volume of the ball).
So, the height of the water is equal to the volume of the ball (381.7 cm^3) divided by the base area of the tin (πr^2).
Substituting the values into the formula, we get:
V = π(6^2)h = 381.7 cm^3
h = 381.7 / (36π) ≈ 3.4 cm (rounded to one decimal place).
3. Calculate how far the water level falls:
When the ball is removed, the water level will decrease by the height of the ball, which is equal to its diameter (9 cm).
Therefore, the water level falls by 9 cm.
To determine how far the water level falls when the ball is removed, we need to calculate the volume of the ball and the volume of water in the can.
1. Calculate the volume of the ball:
The volume of a sphere can be calculated using the formula: V = (4/3) * π * r^3, where r is the radius of the ball.
Given that the diameter of the ball is 9cm, the radius (r) is half the diameter, which is 4.5cm.
Let's plug in the values and calculate the volume of the ball: V_ball = (4/3) * π * (4.5cm)^3.
2. Calculate the volume of water in the can:
The volume of a cylinder is given by the formula: V = π * r^2 * h, where r is the radius of the base and h is the height of the cylinder (water level in this case).
Given that the diameter of the can is 12cm, the radius (r) is half the diameter, which is 6cm.
The height (water level) is not given, so we will label it as 'h'.
3. Substitute the known values and labels into the equation and solve for 'h':
V_water = π * (6cm)^2 * h
4. Substitute the value of the volume of the ball (V_ball) into the equation for V_water:
V_water = V_can - V_ball
Now, to find how far the water level falls, we need to determine the difference in the water levels before and after removing the ball.
Please provide the height label (h) for the original water level in the can, and I can assist you in calculating the difference.