Question

(7) The cost function for coffee at a coffee house is given by C(x) = 3x + 160. The coffee sells for $7 per pound. Find the following: (a) Find the breakeven point: sold: (b) Find the profit when 32 pounds of coffee are

Answer 1

a)

As given we have C(x) = 3x + 160 and coffee sells for $7 hence we can say that R(x) = 7x

we know that break even point is the point where C(x) = R(x)

Hence,

$\dpi{100} \small 3x+160=7x$

$\dpi{100} \small 4x=160$

$\dpi{100} \small x=40$

Hence we can say that 44 pounds of coffee must have to sell to have a break even

b)

we have to find the profit when 32 pounds of coffee are sold

we have,

C(x) = 3x + 160 and R(x) = 7x

we know that profit function is given by P(x) = R(x) - C(x)

Hence,

$\dpi{100} \small P(x) = 7x-(3x+160)$

$\dpi{100} \small P(x) = 7x-3x-160$

$\dpi{100} \small P(x) = 4x-160$

Put x = 32 we can say that,

$\dpi{100} \small P(32) = 4(32)-160$

$\dpi{100} \small P(32) = 128-160$

$\dpi{100} \small P(32) = -32$

minus sign indicates the loss hence we can say that there is a loss of 32 dollars by selling 32 pounds of coffee.

we have to find the profit when 32 pounds of coffee are sold but as 32 < 40(break even point) we have a loss of 32 dollars.