COnsider a box of tennis balls where as many balls are stacked across the bottom as possible. There are six balls on the bottom of the box.

Question

Please help me.

 

You have a larger box of tennis balls. The dimensions of the box are 30cm long, 30cm wide and 30cm high. The radius of each ball is 3cm

 

a) If you stack the balls as close together as possible, how many will be used to cover the bottom of the box?

 

b) How many rows of balls will be needed to fill the box?

 

c) How many balls will fit in the box?

 

d) calculate the volume of the box.

 

e)Calculate the volume of each ball. correct to 1 dec.

 

f) Calculate the volume of the balls used to fill the box, correct to 1 dec place.

 

g) Calculate the volume of the box NOT taken up by tennis balls, correct to 1 dec place.

 

Thanks so much for your help!!!

Answer 1

Imagine a line of balls that are 3 cm wide. How many of the balls fit into 30 cm? Not to scale: x x x Now imagine that you have that many balls lined up next to each other to fill a 30 x 30 cm space. x x x x x x x x x Now add stacks of these until you get to the height of 30 cm. Add up the number of balls The volume of the box is length*width*height The volume of a sphere is (4/3) * pi * r^3 The total volume of balls is equal to the number of balls * the volume of one ball The volume of the box that is not filled with balls is equal to the volume of the box minus the volume of the balls

Answer 2

Hi Deano,

 

You'll need your trusty calculator.

 

The problem is doing everything in cm. So the box is 30 X 30 X 30. Each ball's radius is 3 cm, so its diameter is 6 cm.

 

a. If you pack the balls as close as you can, they'll be in rows and columns on the floor of the box. Each ball is 6 cm across. The box is 30 cm wide. You divide the width of the box by the width of 1 ball, and you get how many balls can fit across, width-wise.

 

The box is also 30 cm long, so you can fit (30 / 6) rows of balls to fill up the floor of the box. Each row of balls has (30 / 6) balls in it, so you wind up with (30 / 6) ROWS X (30 / 6) BALLS PER ROW balls on the floor of the box.

 

b. You need (30 / 6) rows of (30 / 6) balls.

 

c. If you add more rows of balls on top of the bottom layer, you'll get the same number of balls on the second layer - (30 / 6) X (30 / 6) balls on the second layer. So the first 2 layers together have 2 X (30 / 6) X (30 / 6). Since the box is 30 cm high, you can fit (30 X 6) layers of balls in the box. So you'd have (30 / 6) layers X (30 / 6) rows X (30 / 6) balls per row, filling up the box.

 

d. Since the box is 30 X 30 X 30 cm, the volume is 30 cubed (30 to the 3rd power).

 

e. The volume of each ball is 4/3 X pi X (r X r X r) = 4/3 X pi X (3 X 3 X 3). You can simplify that a little: 4/3 X 3 = 4, so you get 4 X pi X 3 X 3. Calculate that out and you get the volume of one ball. {(4 X 3.1416 X 9), or (36 X 3.1416).}

 

f. Now you multiply that volume of one ball times the number of balls filling up the box to get the total ball volume.

 

g. Subtract the ball volume from the box volume. That leaves the non-ball-volume in the box, all the spaces between the balls.

 

Hope that helped.

 

-- Bruce