Tennis balls are stacked four high in a rectangular prism package. The diameter of one ball is 6.5 cm.
a) What is the minimum amount of material needed to make the box?
b) Determine the amount of empty space in the rectangular prism.
no way to tell. All we know is that the box is 4*6.5 = 26 cm high
Now, if the box is a cube, then its area is 6*26^2
the volume of the 64 balls is 64 * π/6 * 6.5^3
subtract that from the volume of the box to get the empty space.
To find the answers to these questions, we need to calculate the dimensions of the rectangular prism first.
Let's start with the height of the rectangular prism. Since the tennis balls are stacked four high, we multiply the diameter of one ball by four: 6.5 cm * 4 = 26 cm.
Next, we need to determine the length and width of the prism. Since the balls are tightly packed, the length and width of the prism will be equal to the diameter of one ball.
Therefore, the dimensions of the rectangular prism are:
Height = 26 cm
Length = 6.5 cm
Width = 6.5 cm
a) To find the minimum amount of material needed to make the box, we need to calculate the surface area of the prism. The surface area is the sum of the areas of all six faces of the box.
To calculate the surface area, we use the formula:
Surface Area = 2*(Length*Width + Length*Height + Width*Height)
Replacing the values with the dimensions of the prism:
Surface Area = 2*(6.5 cm * 6.5 cm + 6.5 cm * 26 cm + 6.5 cm * 26 cm)
Simplifying the expression:
Surface Area = 2*(42.25 cm² + 169 cm² + 169 cm²)
Surface Area = 2*(380.5 cm²)
Surface Area = 761 cm²
Therefore, the minimum amount of material needed to make the box is 761 cm².
b) To determine the amount of empty space (volume) in the rectangular prism, we need to subtract the volume of the tennis balls from the volume of the prism.
The volume of the rectangular prism can be calculated using the formula:
Volume = Length * Width * Height
Substituting the values of the dimensions of the prism:
Volume = 6.5 cm * 6.5 cm * 26 cm
Volume = 1119.5 cm³
Now, let's calculate the volume of one tennis ball using its diameter:
Radius = Diameter / 2 = 6.5 cm / 2 = 3.25 cm
Volume of one ball = (4/3) * π * (radius)^3
Volume of one ball = (4/3) * π * (3.25 cm)^3
Volume of one ball ≈ 179.675 cm³ (rounded to three decimal places)
Since there are four tennis balls stacked in the prism, the total volume of the balls inside the prism is:
Total volume of balls = Volume of one ball * 4
Total volume of balls ≈ 718.7 cm³ (rounded to one decimal place)
To find the amount of empty space, we subtract the volume of the balls from the volume of the prism:
Empty space = Volume of prism - Total volume of balls
Empty space = 1119.5 cm³ - 718.7 cm³
Empty space ≈ 400.8 cm³ (rounded to one decimal place)
Therefore, the amount of empty space in the rectangular prism is approximately 400.8 cm³.