a can of tennis balls consists of three spheres stacked vertically inside a cylinder the balls just Touch the top, bottom, and sides of the cylinder. A diameter of each ball is 6.67 cm. find The Following. use 3.14 for pie and round to the nearest hundredth
height of one ball is 6.67 (its diameter)
height of 3 balls is therefore 20cm
if you want volume of all 3 balls, that'd be 3(1/6 πd^3)
where d is the diameter.
volume of cylinder is just
πr^2 h = π/4 d^2 h = 3π/4 d^3 = 3π/4 (20/3)^3 = 200π/9
Or maybe something else, since you didn't ask a question.
To find the following measures, let's break it down step-by-step:
1. Find the height of the cylinder:
- Since each spherical ball touches both the top and bottom of the cylinder, the height of the cylinder is equal to the diameter of a single ball.
- Given the diameter of each ball as 6.67 cm, the height of the cylinder is also 6.67 cm.
2. Find the radius of the cylinder:
- Since the balls touch the sides of the cylinder, the radius of the cylinder is equal to half the diameter of a ball.
- Given the diameter of each ball as 6.67 cm, the radius of the cylinder is 6.67 cm / 2 = 3.335 cm.
3. Find the volume of the cylindrical container:
- The formula for the volume of a cylinder is V = π * r^2 * h, where π (pi) is approximately 3.14, r is the radius of the cylinder, and h is the height of the cylinder.
- Plugging in the values, V = 3.14 * (3.335 cm)^2 * 6.67 cm.
- Calculate the volume and round it to the nearest hundredth.
4. Find the total volume of the three spheres:
- 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, V = (4/3) * 3.14 * (3.335 cm)^3.
- Calculate the volume and round it to the nearest hundredth.
5. Find the ratio of the total volume of the spheres to the volume of the cylindrical container:
- Divide the total volume of the spheres by the volume of the cylindrical container.
- Round the ratio to the nearest hundredth.
To find the following measurements, we'll need to calculate them step by step:
1. The radius of each ball:
The diameter of each ball is given as 6.67 cm. The radius is half of the diameter, so the radius of each ball is 6.67 cm divided by 2, which is equal to 3.335 cm.
2. The height of the cylinder:
Since the spheres touch the top, bottom, and sides of the cylinder, the height of the cylinder is equal to the combined diameters of two balls.
The diameter of a ball is 6.67 cm, so the combined diameters of two balls is 6.67 cm + 6.67 cm = 13.34 cm.
To find the height, we divide the combined diameters by 2 to get the radius of the cylinder:
13.34 cm / 2 = 6.67 cm.
3. The volume of the cylinder:
The formula for the volume of a cylinder is V = πr²h, where V is the volume, π (pi) is approximately 3.14, r is the radius, and h is the height.
Plugging in the values, we have:
V = 3.14 × (6.67 cm)² × 6.67 cm.
Calculating this expression, we find:
V ≈ 3.14 × (44.3889 cm²) × 6.67 cm ≈ 3.14 × 296.1353 cm³ ≈ 930.19 cm³.
4. The surface area of the cylinder:
The formula for the surface area of a cylinder is A = 2πrh + 2πr².
Plugging in the values, we have:
A = 2 × 3.14 × (6.67 cm) × 6.67 cm + 2 × 3.14 × (6.67 cm)².
Calculating this expression, we find:
A ≈ 2 × 3.14 × 44.3889 cm² + 2 × 3.14 × 296.1353 cm² ≈ 2785.71 cm².
Therefore, the measurements are as follows:
- Volume of the cylinder: approximately 930.19 cm³.
- Surface area of the cylinder: approximately 2785.71 cm².