A company distributes college logo sweatshirts and sells them for \( \$ 45 \) each. The total cost function is linear, and the total cost for 70 sweatshirts is \( \$ 4136 \), whereas the total cost for 220 sweatshirts is \( \$ 5785 \). (a) Write the equation for the revenue function \( R(x) \). \( R(x)=\square \) (b) Write the equation for the total cost function \( C(x) \). \( C(x)=\square \) (c) Find the break-even quantity. \( x=\square \)

To solve the problem, we will go through each part step by step. ### (a) Write the equation for the revenue function \( R(x) \). The revenue function \( R(x) \) is calculated by multiplying the number of sweatshirts sold \( x \) by the price per sweatshirt. Given that each sweatshirt sells for \( \$ 45 \), the revenue function is: \[ R(x) = 45x \] ### (b) Write the equation for the total cost function \( C(x) \). The total cost function is linear, which can be expressed in the form: \[ C(x) = mx + b \] where \( m \) is the variable cost per sweatshirt and \( b \) is the fixed cost. We have two points to find the slope \( m \) and the intercept \( b \): 1. For 70 sweatshirts: \( C(70) = 4136 \) 2. For 220 sweatshirts: \( C(220) = 5785 \) First, we can find the slope \( m \): \[ m = \frac{C(220) - C(70)}{220 - 70} = \frac{5785 - 4136}{220 - 70} = \frac{1649}{150} \approx 10.9933 \] Now, we can use one of the points to find \( b \). Let's use the point \( (70, 4136) \): \[ C(70) = m \cdot 70 + b \] \[ 4136 = 10.9933 \cdot 70 + b \] \[ 4136 = 769.531 + b \] \[ b = 4136 - 769.531 \approx 3366.469 \] Thus, the total cost function is: \[ C(x) = 10.9933x + 3366.469 \] ### (c) Find the break-even quantity. The break-even point occurs when revenue equals cost: \[ R(x) = C(x) \] \[ 45x = 10.9933x + 3366.469 \] Now, we can solve for \( x \): \[ 45x - 10.9933x = 3366.469 \] \[ 34.0067x = 3366.469 \] \[ x \approx \frac{3366.469}{34.0067} \approx 98.9 \] Since \( x \) must be a whole number, we round to the nearest whole number: \[ x \approx 99 \] ### Summary of Answers: (a) \( R(x) = 45x \) (b) \( C(x) = 10.9933x + 3366.469 \) (c) \( x = 99 \)