Question: 17. At a soccer game, a group of 100 volunteers in the stadium will hold up cards to form pictures…

17. At a soccer game, a group of 100 volunteers in the stadium will hold up cards to form pictures. Each display uses 100 square cards. They are most effective when the volunteers sit in a rectangular arrangement. What rectangular seating arrangements are possible? Which arrangements would you choose? Why?

Solution

To solve the problem, we need to find the rectangular seating arrangements possible with 100 volunteers. Each arrangement corresponds to the factors of 100 since the number of volunteers corresponds to the area of the rectangle (length × width = 100). First, identify all pairs of whole numbers whose product is 100: \[ \begin{align*} 1 \times 100 &= 100 \\ 2 \times 50 &= 100 \\ 4 \times 25 &= 100 \\ 5 \times 20 &= 100 \\ 10 \times 10 &= 100 \\ \end{align*} \] Next, identify the rectangular seating arrangements (length × width): 1. \(1 \times 100\) 2. \(2 \times 50\) 3. \(4 \times 25\) 4. \(5 \times 20\) 5. \(10 \times 10\) The arrangement you would choose depends on visibility and aesthetic preference: - The \(10 \times 10\) arrangement is the most balanced and compact, offering good visibility for spectators. - The choice will depend on the stadium configuration and desired effect, but \(10 \times 10\) is generally preferred for symmetry and visibility. Thus, an arrangement like \(10 \times 10\) could be most effective for a balanced display.