The Great Soccer Ball Giveaway

Like the How Fast Are You Moving? puzzle, this challenge is a Fermi question.

The Seedonia National Soccer Team (the Seedies) made it to the final 16 in World Cup Soccer for the first time, and the team wants to reward its fans for their support and encouragement. At the same time they want to conduct a major fund-raiser for their next World Cup attempt. They also want to inaugurate their new stadium, one of the largest in the world.

Eventually the Seedies public relations department came up with a promotion that would bring the fans into the stadium, raise some much needed income, and reward their loyal fans, all at the same time. They called their promotion The Great Soccer Ball Giveaway!

Here’s the idea. The Seedies will play an exhibition game against East Seedonia, a famous rivalry. Seedonia’s sporting goods manufacturers have agreed to donate a large number of soccer balls—enough to fill the entire playing area. At the end of the game, trucks will drive onto the field and deliver all the soccer balls. The team members will arrange them to fill the entire playing area.

At a signal from the team manager, the fans will be invited to come down onto the field and take one of the soccer balls.

Here are the questions that the public relations department has to answer. Can you help them?

How many soccer balls are needed to fill the playing area of Seedonia Stadium?

Will there be enough of them so that every fan in the stadium can have one?

How many trucks will the team have to rent to deliver all the soccer balls?

You’ll need some information about soccer balls, soccer fields, and truck sizes in order to do this. Some of this information can be found on the Web site of the international soccer organization, FIFA [1] (Federation Internationale de Football Association). You can learn about the sizes of stadiums around the world at The World Stadiums Web site [2].

For the rest, you’re on your own.

Hint: Make a rough estimate first. Then when you use the actual numbers, you’ll have a good check on the accuracy of your calculations.

Background

Fermi questions

This puzzle and How Fast Are You Moving? [3] are examples of “Fermi questions”, named in honor of the Nobel laureate and nuclear physicist Enrico Fermi. Fermi used to give his students problems that involved large numbers, estimates and approximations. He expected his students to be able to simplify the numbers and do all their calculations on a small piece of paper. They are sometimes called “back of the envelope” problems. You can find many examples on the Web if you search for “Fermi questions.”

Fermi questions involve making many assumptions, approximations, and simplifications to try to get a reasonable answer quickly. Your answer doesn’t have to be exactly correct—maybe there is no exact answer, because the problem is vague or conditions can vary. A Fermi question does not give you all the information you need; you need to look things up and/or make assumptions.

Course:

Math [4]

Result/Solution(s)

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 m³. (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 m³ 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 m³. Let’s assume we’re renting 20 m³ 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.

Math Puzzle [5]