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Solution: Math Puzzle: The Great Soccer Ball Giveaway

# Solution: The Great Soccer Ball Giveaway Math Puzzle

### Make a Rough Estimate

Let’s try a rough estimate for the number of soccer balls that are to be spread onto the playing area of a soccer field.

Think about a soccer ball; imagine its size. It’s probably about 25 cm (10 in) in diameter. If we put four of them next to each other they’d equal a length of 1 m (3.3 ft).

Sixteen soccer balls, each 25 cm in diameter, would exactly fill 1 sq m (11 sq ft).

Think about a soccer field. It’s probably about 100 m (330 ft) long and 50 m (165 ft) wide. That’s 5,000 m (16,400 ft). Our first estimate then is that the field will hold 16 x 5,000, or 80,000, soccer balls. That would be enough for all the fans in a **very** large stadium. Can you imagine a soccer stadium that large?

### Check the Estimate

Let’s check our estimate of how many balls fit on a soccer field by using some real data. The international soccer federation, FIFA, oversees the World Cup competition and other international matches. Here’s some of the data from their Web site:

An official soccer ball must be between 68 and 70 cm (27 and 28 in) in circumference.

For international play a soccer field can be between 100 and 110 m (330 and 360 ft) long and from 64 to 75 m (210 and 246 ft) across.

We can use the classic formula for the circumference of a circle, C = π D, to find the diameter of the largest sized soccer ball, 70 / 3.14 = 22.29 cm. Let’s take 22 cm (9 in) as a good approximation.

To find the number of balls that would fit in the smallest size soccer field, divide 100 m by 0.22 m (22 centimeters converted to meters) to figure out that 454.55, or approximately 450 balls, can fit along the length of the field.

Across the field we have 64 / 0.22 = 290.9, or about 29.

So we have a total of 450 x 290 = 130,500 soccer balls.

This is more than our estimate (80,000) but still fairly close (we estimated that a soccer ball had a diameter of 25 cm, which was a little too large, and that a field was 100 x 50 m, which turned out to be too small).

### How Many Fans Can a Stadium Hold?

If you search the Internet for world soccer stadiums, you’ll find that most soccer stadiums hold many fewer than 130,000 people. But some of the largest in major countries are close to that figure. The World Stadiums Web site lists 44 stadiums worldwide that hold 100,000 people or more. Most of these are used for auto racing or horse racing.

How big a stadium would a small country like Seedonia have? You’ll have to use your imagination.

### Getting the Balls to the Stadium

So, how many trucks will it take to carry 130,500 soccer balls? To answer this, we’ll need to consider volume. First, for simplicity’s sake, let’s consider that the balls are stacked directly on top of each other. How many balls, stacked on top of each other would fill 1 cu m? Since each ball has a diameter of 22 cm, each takes up a volume of .22 x .22 x .22 = .0106 m^{3}. (This is *not* the volume of the ball, but of a cube that contains the ball. Imagine that the ball comes packed in a box, 22 cm on a side. Then .0106 m^{3} is the volume of that box.) One cubic meter could hold 94.34 such boxes.

Now we need to know how many soccer balls a truck will hold. You’ll have to check on the sizes of trucks available in your country. A company in western Europe, Europcar, rents trucks with volumes of 10, 12, and 20 m^{3}. Let’s assume we’re renting 20 m^{3} trucks. Each truck will hold approximately 20 / 0.0106 = 1,887 soccer balls. In order to carry this many balls, a truck would have to have exactly the correct dimensions. For example, a truck that is 5 m (16 ft) long and 2 m (7 ft) wide could hold a layer of balls, 22 long x 9 across. This leaves a little bit of space that is unused.

In order to transport 130,500 balls, we’ll need 130,500 ÷ 1887 = 69.15, for a total of 70 trucks! It might be a problem to find that many trucks to rent. If we used larger trucks—say tractor-trailers—we wouldn’t need as many.