Question

problem 3: two different community centres organize end of year fairs for their communities.

they each charge an entrance fee and a cost per game. the information below shows how much fair a and fair b each charge if someone were to play a certain number of games.

fair a
\

$$\begin{array}{|c|c|} \\hline \\text{number of games played} & \\text{total cost (\\$)} \\\\ \\hline 4 & 53 \\\\ \\hline 6 & 54.50 \\\\ \\hline 8 & 56 \\\\ \\hline \\end{array}$$

fair b
graph showing points at (10, 36) and (20, 52)

a) determine an equation for the total cost (c) for each of the fairs, for a certain number of games played, n. show your work. (4 marks - 2 each)

fair a: \underline{\qquad\qquad\qquad\qquad} fair b: \underline{\qquad\qquad\qquad\qquad}

b) for how many games played will both fairs cost the same money? solve algebraically. (2 marks)

Explanation:

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<reasoning>

Find the linear equation for Fair A

\[

$$\begin{aligned} m_A &= \frac{54.50 - 53}{6 - 4} = \frac{1.50}{2} = 0.75 \\ C_A - 53 &= 0.75(n - 4) \\ C_A &= 0.75n + 50 \end{aligned}$$

\]

Find the linear equation for Fair B

\[

$$\begin{aligned} (n_1, C_1) &= (10, 35), \quad (n_2, C_2) = (20, 50) \\ m_B &= \frac{50 - 35}{20 - 10} = \frac{15}{10} = 1.5 \\ C_B - 35 &= 1.5(n - 10) \\ C_B &= 1.5n + 20 \end{aligned}$$

\]

Solve for the intersection point algebraically

\[

$$\begin{aligned} C_A &= C_B \\ 0.75n + 50 &= 1.5n + 20 \\ 30 &= 0.75n \\ n &= 40 \end{aligned}$$

\]
</reasoning>

<answer>

Question a

Fair A: \(C = 0.75n + 50\)

Fair B: \(C = 1.5n + 20\)

Question b

\(n = 40\) games
</answer>

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Answer:

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<reasoning>

Find the linear equation for Fair A

\[

$$\begin{aligned} m_A &= \frac{54.50 - 53}{6 - 4} = \frac{1.50}{2} = 0.75 \\ C_A - 53 &= 0.75(n - 4) \\ C_A &= 0.75n + 50 \end{aligned}$$

\]

Find the linear equation for Fair B

\[

$$\begin{aligned} (n_1, C_1) &= (10, 35), \quad (n_2, C_2) = (20, 50) \\ m_B &= \frac{50 - 35}{20 - 10} = \frac{15}{10} = 1.5 \\ C_B - 35 &= 1.5(n - 10) \\ C_B &= 1.5n + 20 \end{aligned}$$

\]

Solve for the intersection point algebraically

\[

$$\begin{aligned} C_A &= C_B \\ 0.75n + 50 &= 1.5n + 20 \\ 30 &= 0.75n \\ n &= 40 \end{aligned}$$

\]
</reasoning>

<answer>

Question a

Fair A: \(C = 0.75n + 50\)

Fair B: \(C = 1.5n + 20\)

Question b

\(n = 40\) games
</answer>

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