what is the maximum weekly profit

Question

A local sandwich store has a fixed weekly cost of $525 and variable costs for making a roast beef sandwich are $0.55. let x represent the number of roast beef sandwiches made and sold each week. the function R (x)=-0.001x^2+3x describes the money, in dollars that the shop takes in each week from the sale f x roast beef sandwiches.

Answer 1

Cost Function:

C(x) = 525 + .55x

Revenue Function:

R(x) = -.001x^2 + 3x -----> given

Profit Function: (revenue function minus cost function)

P(x) = -.001x^2 + 3x - (525 + .55x)

= -.001x^2 + 3x - 525 - 0.55x

P(x) = -.001x^2 + 2.45x - 525

Parabola = negative leading coefficient means it will open downward. This means your vertex will be your maximum. To find your vertex -b/2a.

-2.45/(2*-.001)

=2.45/.002

=1225 sandwiches

Plug this into your profit formula.

-.001(1225)^2 + 2.45 (1225) - 525

-1500.625 + 3001.25 -525

975.625

Because we are talking money we round down to 2 decimal places. $975.62.

The sandwich shop should make and sell per week 1,225 sandwiches for a profit of $975.62.

Answer 2

 

COST FUNCTION:

 

     Fixed cost = 525

     Variable cost = .55x. where x is the number of sandwiches made

 

     C(x) = .55x + 525

 

REVENUE FUNCTION: (given)

    

     R(x) = -.001x² + 4x

 

FINDING MAXIMUM VALUE:

 

     The revenue function is a quadratic equation.  It opens downward ... the lead coefficient is negative.

     The x coordinate occurs at -b/2a

 

      a = -.001  (coefficient of the x² term)

      b = 4        (coefficient of the x term)

 

     -b/2a = -4/(-.001) = 4000

 

      The maximum weekly profit occurs when 4000 sandwiches are made and sold.

      The maximum weekly revenue is found by plugging 4000 into the revenue function.

 

      R(4000) = -.001(4000)² + 4(4000)

                 = 32,000

 

      We must subtract the cost of making those sandwiches, to obtain the profit.

 

      C(4000) = .55(4000) + 525

                  = 2725

 

PROFIT:

     

       Profit = Revenue - cost

               = 32,000 - 2725

               = 29,275

 

The maximum weekly profit is $29,275