Tennis balls are stacked four high in a rectangular prism package. The diameter of one ball is 6.5cm. a) Calculate the volume of the rectangular...
package. b) What is the minimum amount of material needed to make the box? c) Determine the amount of empty space in the rectangular prism package. d) what assumptions have you made. Thank you for any help.
The diameter of the balls =6.5cm. Each rectangular prism shaped pack contains 4 tennis balls kept high one over the other . So house the arrangement the required(inner dimension of the pack = 6.5 cm length , 6.5 cm width and 6.5*4 = 26cm high.
a) the volume of the rectangular prism = length*length*height = 6.5^2*(6.5*4) =1098.5 cm^3.
b)The (minimum )amount of material to make the fixed shape is (prportional) equal to the area of the 6 surface of the rectangular box= 2(lw+lh+wh) = 2(3w^2+2w^2), as l=w and h = 2w . So the surface area = 10*6.5^2 = 422.5 sq cm.
The minimum material to cover the 4 balls to keep in the fixed stack shape does not arise.
The amount of empty space in the box = Volume of the rectangular prism - volume of the 4 boxes = 6.5*2*26 - 2*(4/3)(Pi*r^3^2) = 1098.5 - 2*(4/3)pi(6.5/21006.96)^3 = 523.33 cm^3.
d) We assumed that the material is (proportional) equal to the area of the surface .
The tennis balls are spherical in shape and cannot be changed in shape. So we cannot take the material used to make one ball and place it in the rectangular prism package. We need a rectangular prism package that can accommodate each of the balls.
So the height of the package is 4*6.5 = 26 cm. The width of the box is 6.5 cm and the length is also 6.5 cm.
So the volume of the package is: 1098.5 cm^2
The amount of material needed to make the box is 6.5*6.5*2 for the base and the top + 4*6.5*6.5*4= 760.5cm^2
The amount of empty space is (approximately) : 1098.5 - 4*(4/3)*pi*3.25^3=523.3 cm^3