Three balls are packaged in a cylindrical container as shown below. 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?
(This is 10th grade math, please don't make it advanced)
AAAaannndd the bot gets it wrong yet again!
You only figured a cylinder 1 ball high.
The cylinder is 3 balls high, or 39cm
so the volume of the cylinder is
π * 6.5^2 * 39 = 1648π cm^3
each ball has volume 366π cm^3, so the 3 balls occupy 1098.5π cm^3
The balls occupy 1098.5/1648 = 0.665 = 67%
I apologize for my mistake. Thank you for correcting me.
a. To find the volume of the cylinder, we will use the formula:
V_cylinder = π * r^2 * h
Since the balls touch the top and bottom of the cylinder, the height of the cylinder is equal to the diameter of one ball, which is 13 cm.
The radius of the cylinder can be determined by taking half of the diameter of the ball. So, the radius is 13 cm / 2 = 6.5 cm.
Plugging these values into the formula, we have:
V_cylinder = π * (6.5 cm)^2 * 13 cm
V_cylinder ≈ 1338.26 cm³ (rounded to the nearest cubed centimeter)
Therefore, the volume of the cylinder rounded to the nearest cubed centimeter is approximately 1338 cm³.
b. The total volume of the three balls can be calculated by finding the volume of a single ball and then multiplying it by 3.
The formula for the volume of a sphere is:
V_sphere = (4/3) * π * r^3
Since the diameter of the ball is 13 cm, the radius is 6.5 cm. Plugging this into the formula, we have:
V_sphere = (4/3) * π * (6.5 cm)^3
V_sphere ≈ 1490.34 cm³ (rounded to the nearest cubed centimeter)
Since we have three balls, the total volume of the three balls is:
3 * V_sphere = 3 * 1490.34 cm³ ≈ 4470.10 cm³ (rounded to the nearest cubed centimeter)
Therefore, the total volume of the three balls rounded to the nearest cubed centimeter is approximately 4470 cm³.
c. To find the percentage of the volume of the container occupied by the three balls, we divide the total volume of the three balls by the volume of the cylinder and multiply by 100.
Percent occupied = (Total volume of balls / Volume of cylinder) * 100
Plugging in the values, we have:
Percent occupied = (4470 cm³ / 1338 cm³) * 100
Percent occupied ≈ 334.69 %
Therefore, the three balls occupy approximately 334.69% of the volume of the container.
To solve this problem, we need to calculate the volume of the cylindrical container and the total volume of the three balls, then determine the percentage of the container's volume occupied by the balls.
a. Volume of the cylinder:
The volume of a cylinder is given by the formula V = πr^2h, where r is the radius of the cylinder's base and h is the height. In this case, the height of the cylinder is equal to the diameter of one ball, which is 13 cm. The radius can be found by dividing the diameter by 2, so the radius is 13 cm / 2 = 6.5 cm.
Using the formula, the volume of the cylinder is V = π(6.5 cm)^2(13 cm) = 4395.44 cm^3. Rounded to the nearest cubic centimeter, the volume of the cylinder is 4395 cm^3.
b. Total volume of the three balls:
The volume of a sphere is given by the formula V = (4/3)πr^3, where r is the radius of the sphere. In this case, the diameter of each ball is 13 cm, so the radius is 13 cm / 2 = 6.5 cm.
Since there are three balls, the total volume is 3 * (4/3)π(6.5 cm)^3 = 843.03 cm^3. Rounded to the nearest cubic centimeter, the total volume of the three balls is 843 cm^3.
c. Percentage of the volume occupied by the balls:
To find the percentage, we need to divide the total volume of the three balls by the volume of the cylinder, and then multiply by 100.
Percentage = (Total volume of balls / Volume of cylinder) * 100
= (843 cm^3 / 4395 cm^3) * 100
= 19.18%
Therefore, the three balls occupy approximately 19.18% of the volume of the cylindrical container.
a. The diameter of each ball is 13 cm, so the radius is 6.5 cm. The height of the cylinder is equal to the diameter of the ball, so it is also 13 cm. The formula for the volume of a cylinder is V = πr^2h, where π is approximately equal to 3.14. Plugging in the values, we get:
V = 3.14 × 6.5^2 × 13
V ≈ 2222 cm^3 (rounded to the nearest cubed centimeter)
b. The formula for the volume of a sphere is V = (4/3)πr^3. Plugging in the values, we get:
V = (4/3) × 3.14 × 6.5^3
V ≈ 1796 cm^3 (rounded to the nearest cubed centimeter)
Since we have three balls, the total volume is:
1796 × 3 = 5388 cm^3 (rounded to the nearest cubed centimeter)
c. To find the percent of the volume of the container occupied by the three balls, we need to divide the total volume of the balls by the volume of the cylinder and multiply by 100. So:
percent = (5388 / 2222) × 100
percent ≈ 242.7% (rounded to the nearest tenth)
This doesn't make sense as a percentage, as it suggests that the three balls take up more than twice the volume of the container. This is because the balls are touching each other and the sides of the container, so they overlap and leave empty space. Therefore, we should conclude that the balls occupy more than 100% of the volume of the container, but less than 200%.