A boutique owner sells handcrafted scarves. The fixed monthly costs are $500 , and the variable cost per scarf is $10 The boutique's revenue is $0 when 0 scarves are sold and $0 when 100 scarves are sold due to market saturation. The maximum revenue occurs when 50 scarves 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 scarves sold?
Answer
A
cap c times x is equal to 500 x plus 10 cap r times x is equal to negative x times open paren x minus 50 close paren
B
cap c times x is equal to 500 x plus 10 cap r times x is equal to negative x times open paren x minus 100 close paren
C
cap c times x is equal to 10 x plus 500 cap r times x is equal to negative x times open paren x minus 100 close paren
D
cap c times x is equal to 10 x plus 500 cap r times x is equal to negative x times open paren x minus 50 close paren
Explanation
The cost function adds the $500 fixed cost to $10 for each scarf, so C(x) = 10x + 500. Revenue is zero at x = 0 and x = 100, so its factors are x and (x − 100); the negative sign makes the parabola open downward, giving R(x) = −x(x − 100). Its vertex is halfway between the zeros, at x = 50, where revenue is maximal. Therefore, choice C is correct.
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