Question

A sporting goods company is planning to manufacture a commemorative lacrosse ball to demonstrate the importance of lacrosse to Canadian culture. The ball has a diameter of 10 cm. Determine the dimensions of the cylinder which will package the lacrosse ball so that it has a maximum volume and minimum surface area. Calculate the volume and surface area of this cylinder.

a) r=5cm,h=20cm,V=523.6cm^3 ,SA=471.24cm^2
b) r=10cm,h=10cm,V=1000cm^3 ,SA=628.32cm^2
c) r=20cm,h=5cm,V=1570.8cm^3 ,SA=942.48cm^2
d) r=15cm,h=10cm,V=706.5cm^3 ,SA=942.48cm^2

Answer 1

To determine the dimensions of the cylinder packaging the lacrosse ball with maximum volume and minimum surface area, we find the radius and height of the cylinder. The dimensions are a radius of 5 cm and height of 12.72 cm. The volume of the cylinder is 1000 cm³, and the surface area is 628.32 cm².

Step-by-step explanation:

In order to determine the dimensions of the cylinder that will package the lacrosse ball and have a maximum volume and minimum surface area, we need to find the radius and height of the cylinder.

1. First, we know that the diameter of the lacrosse ball is 10 cm, so the radius is half of that, which is 5 cm.
2. To find the height of the cylinder, we can use the formula for the volume of a cylinder: V = πr²h.
3. Given the volume of the cylinder is 1000 cm³, we can plug in the values for V and r to solve for h: 1000 = 3.142 × (5 cm)² × h.
4. By rearranging the formula, we get h = 1000 / (3.142 × 25) = 12.72 cm.

Therefore, the dimensions of the cylinder are: radius = 5 cm and height = 12.72 cm.

To calculate the volume and surface area of the cylinder:

1. The volume of the cylinder is given by the formula V = πr²h. Substituting the values, we get V = 3.142 × (5 cm)² × 12.72 cm = 1000 cm³.
2. The surface area of the cylinder is given by the formula SA = 2πrh + 2πr². Substituting the values, we get SA = 2 × 3.142 × 5 cm × 12.72 cm + 2 × 3.142 × (5 cm)² = 628.32 cm².