How many hexagons could Don build with 61 toothpicks if he linked them together like this chain?Don could build a chain with  hexagons.Answer

To determine how many hexagons Don can build with 61 toothpicks linked together, we need to analyze how many toothpicks are needed to create a chain of hexagons.

Step 1: Break Down the Problem

1. Understand that each hexagon has 6 sides (and thus requires 6 toothpicks).
2. When linking hexagons, each adjacent hexagon will share a side with the previous hexagon.

Step 2: Relevant Concepts

1. For $n$ hexagons, the total number of toothpicks can be calculated as follows:

   • The first hexagon uses 6 toothpicks.
   • Each additional hexagon uses 5 additional toothpicks (since it shares one with the previous hexagon).

   Therefore, the formula for the number of toothpicks $T$ needed for $n$ hexagons is: $T = 6 + 5(n - 1)$ Simplifying this gives: $T = 5n + 1$

Step 3: Analyze and Calculate

1. Set up the equation using the total number of toothpicks available (61): $5n + 1 = 61$
2. Subtract 1 from both sides: $5n = 60$
3. Divide both sides by 5: $n = 12$

Step 4: Verify and Summarize

1. Check the calculation by substituting $n$ back into the original formula $T = 5n + 1$: $T = 5(12) + 1 = 60 + 1 = 61$ This confirms that the calculation is correct.

Final Answer

Don could build 12 hexagons with 61 toothpicks.