A coffee shop has fixed monthly expenses of $2,000 and a variable cost of $2 per cup of coffee sold. The revenue is $0 when 0 cups are sold and $0 when 400 cups are sold due to capacity limits. The maximum revenue occurs when 200 cups are sold. Which system of equations represents the linear cost function C(x) and the quadratic revenue function R(x) , where x represents the number of cups of coffee sold?
Answer
A
cap c times x is equal to 2 comma 000 x plus 2 cap r times x is equal to negative x times open paren x minus 400 close paren
B
cap c times x is equal to 2 x plus 2 comma 000 cap r times x is equal to negative x times open paren x minus 400 close paren
C
cap c times x is equal to 2 x plus 2 comma 000 cap r times x is equal to negative x times open paren x minus 200 close paren
D
cap c times x is equal to 2 comma 000 x plus 2 cap r times x is equal to negative x times open paren x minus 200 close paren
Explanation
The cost function adds the fixed monthly expense to the variable cost for each cup: C(x) = 2,000 + 2x. Revenue is zero at x = 0 and x = 400, so its quadratic factors are x and (x − 400). A negative leading coefficient makes the parabola open downward, with its maximum halfway between the zeros, at 200 cups. Thus the matching system is C(x) = 2x + 2,000 and R(x) = −x(x − 400), choice B.
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