Three balls are packaged in a cylindrical container. The balls just touch the top, bottom, and sides of the cylinder. The diameter
of each ball is 13 cm.
a. What is the volume of the cylinder rounded to the nearest cubed centimeter?
b. What is the total volume of the three balls rounded to the nearest cubed centimeter ?
c. What percent of the volume of the container is occupied by the three balls?
Thank You :)
To solve this problem, we need to calculate the volume of the cylinder and the volume of three balls. Then, we can find the percentage of the volume the balls occupy in the container.
a. Volume of the cylinder:
The diameter of each ball is 13 cm, so the radius of each ball is half of the diameter, which is 6.5 cm. Since the balls just touch the top, bottom, and sides of the cylinder, the height of the cylinder is equal to the diameter of the ball, which is 13 cm.
Using the formula for the volume of a cylinder: V = πr^2h, we can calculate the volume of the cylinder:
V_cylinder = π(6.5^2)(13)
V_cylinder ≈ 4,178.13 cm^3
Rounded to the nearest cubed centimeter, the volume of the cylinder is approximately 4,178 cm^3.
b. Volume of three balls:
The diameter of each ball is 13 cm, so the radius of each ball is 6.5 cm.
Using the formula for the volume of a sphere: V = (4/3)πr^3, we can calculate the volume of each ball:
V_ball = (4/3)π(6.5^3)
V_ball ≈ 2243.67 cm^3
Since there are three balls, the total volume of the three balls is:
V_total = 3(V_ball)
V_total ≈ 3(2243.67)
V_total ≈ 6731.01 cm^3
Rounded to the nearest cubed centimeter, the total volume of the three balls is approximately 6731 cm^3.
c. Percentage of the volume occupied by the three balls:
To find the percentage of the volume occupied by the three balls, we can divide the volume of the three balls by the volume of the cylinder and multiply by 100.
Percentage occupied = (V_total/V_cylinder) * 100
Percentage occupied = (6731/4178.13) * 100
Percentage occupied ≈ 160.96%
Therefore, approximately 160.96% of the volume of the container is occupied by the three balls.
To solve this problem, let's follow these steps:
Step 1: Find the volume of a single ball.
We are given that the diameter of each ball is 13 cm. The radius of a ball is half its diameter, so the radius of each ball is 13 cm / 2 = 6.5 cm. The formula for the volume of a sphere is V = (4/3) * π * r^3, where r is the radius of the sphere. Plugging in the values, we get:
V_ball = (4/3) * π * (6.5 cm)^3
Step 2: Find the volume of the cylindrical container.
The cylindrical container touches the top, bottom, and sides of the balls. This means the height of the cylinder is equal to the diameter of a ball, which is 13 cm. The formula for the volume of a cylinder is V = π * r^2 * h, where r is the radius and h is the height. Since the radius of the cylinder is the same as the radius of the ball, which is 6.5 cm, we can calculate the volume of the cylinder as:
V_cylinder = π * (6.5 cm)^2 * 13 cm
Step 3: Find the volume of three balls.
Since there are three balls, we can simply multiply the volume of a single ball by 3:
V_3balls = 3 * V_ball
Step 4: Calculate the percentage volume occupied by the three balls.
To find the percentage of the container occupied by the three balls, we need to divide the volume of the three balls by the volume of the cylinder, and then multiply by 100. The formula is:
% volume = (V_3balls / V_cylinder) * 100
Now, we can calculate the answers to the questions:
a. Volume of the cylinder:
Using the formula from Step 2, we get:
V_cylinder = π * (6.5 cm)^2 * 13 cm
b. Volume of the three balls:
Using the formula from Step 3, we get:
V_3balls = 3 * V_ball
c. Percentage volume occupied by the three balls:
Using the formula from Step 4, we get:
% volume = (V_3balls / V_cylinder) * 100
Finally, round the results to the nearest cubic centimeter.
height= 13*3cm
diametercan=13cm
volume= PI(d/2)^2 * h
total volume balls= 3*4/3 PI (D/2)^3