Question

The brick oven bakery sells more loaves of bread when it reduces its price but then its profits change. The function y=-100(x-1.75)) squared + 300 models the bakeries profits, in dollars, where x is the price of a load of bread in dollars. The bakery wants to.maximize its profits.
a). What is the domain of this function? Can x ever be negative? Explain.

b). Find the daily profit for selling the bread at $2.00 per loaf.

c). What price should the bakery change to maximize its profits?
d). What is the maximum profit?

Answer 1

Given profit function $y=-100(x-1.75)^2+300$ where y gives profit when x is the price of a load of bread in dollars.

Ans(a):

Domain means what values of x can be taken for given function so that function is defined.

Since x represents the price of a load of bread in dollars which can't be negative in practice.

So domain will be all positive real numbers. We can also write x>0

x can't be negative.

Ans(b):

Plug x=2 into given equation.

$y=-100(2-1.75)^2+300=-100(0.25)^2+300=-100(0.0625)+300=-6.25+300=293.75$

Hence profit when load of bread is sold at $2 is $293.75.

Ans(c):

Given function is quadratic and we know that maximum occurs at the vertex of quadratic function.

compare given quadratic equation with $y=a(x-h)^2+k$, we get:

h=1.75 and k=300

For quadratic equation $y=a(x-h)^2+k$, vertex is given by (h,k)

Hence vertex for given equation is (1.75,300)

Hence bakery should change selling price to $1.75 in order to get maximum profit.

Ans(d):

Maximum profit as per above calculation is $300.