2. (16 pts.) A factory finds that it costs \( \$ 3,600 \) to produce 100 chairs in one day and it costs \( \$ 5,200 \) to produce 300 chairs in one day. (a) Express the daily cost \( C \) as a function of \( x \), where \( x \) is the number of chairs produced. Assume that \( C \) is a lineartion of \( x \). (b) What is the C-intercept of the line and what is its interpretation? (c) What is the slope of this line and what is its interpretation? (e) If the daily cost of producing chairs is \( \$ 5,064 \), use the function to find the number of chairs produced. (d) Use the function to predict the daily cost of producing 470 chairs. (d) (b) ( 10

To determine the daily cost \( C \) as a function of the number of chairs produced \( x \), we need to find the linear equation of the line that passes through the given points. With the data provided, we can derive this linear function. First, let's find the slope \( m \): Given two points, \( (100, 3600) \) and \( (300, 5200) \): \[ m = \frac{C_2 - C_1}{x_2 - x_1} = \frac{5200 - 3600}{300 - 100} = \frac{1600}{200} = 8 \] Now, using point-slope form, we can create the linear cost function: \[ C(x) = mx + b \] We also need to determine \( b \). Using one of the points, for example, \( (100, 3600) \): \[ 3600 = 8(100) + b \implies b = 3600 - 800 = 2800 \] Therefore, the cost function is: \[ C(x) = 8x + 2800 \] Now, let's answer each part of the question: (b) The C-intercept occurs when \( x = 0 \), which gives us \( C(0) = 2800 \). This represents the fixed costs of the factory when no chairs are being produced, meaning it has to spend \( \$ 2,800 \) even without production. (c) The slope of the line is \( 8 \), which means that for each additional chair produced, the daily cost increases by \( \$ 8 \). This increment reflects the variable cost associated with production. (d) To find the number of chairs produced for a daily cost of \( \$ 5,064 \): \[ 5064 = 8x + 2800 \implies 8x = 5064 - 2800 \implies 8x = 2264 \implies x = \frac{2264}{8} = 283 \] Thus, at this cost, the factory produces 283 chairs. (e) To predict the daily cost of producing 470 chairs: \[ C(470) = 8(470) + 2800 = 3760 + 2800 = 6560 \] So, producing 470 chairs will cost the factory \( \$ 6,560 \). In a nutshell, we've derived the cost function, identified the intercept and slope, and solved both production and cost inquiries based on the function! Keep those costs in check, and happy chair-making!