SOLUTION: Mary works a part-time job, selling cards over the telephone. She sells two types of cards, birthday cards and holiday cards. Mary makes $2.00 for each box of birthday cards she se

Question 1099483: Mary works a part-time job, selling cards over the telephone. She sells two types of cards, birthday cards and holiday cards. Mary makes $2.00 for each box of birthday cards she sells, and she makes $2.50 for each box of holiday cards she sells. Mary can work no more than 10 hours (600 minutes) per week. It takes her an average of 15 minutes to sell one box of birthday cards and an average of 20 minutes to sell one box of holiday cards. If Mary can sell no more than 35 boxes of cards in total, how many boxes of each type should she sell to make the most money?

Answer by Theo(13342) (Show Source): You can put this solution on YOUR website!
you can solve this graphically.

let x = number of birthday cards sold.
let y = number of holiday cards sold.

the selling price is 2 dollars for a birthday card and 2.50 for a holiday card.

your objective function is 2x + 2.5y.

this is what you want to maximize.

it takes her 15 minutes to sell a birthday card and 20 minutes to sell a holiday card and the total minutes she has available is 600.

15x + 20y <= 600 is one of her constraints.

she can sell no more than 35 cards.

x + y <= 35 is another of her constraints.

the number cards sold can't be less than 0.

x >= 0 and y >= 0 are another of her constraints.

her objective function is revenue = 2x + 2.5y

her constraints are:

15x + 20y <= 600
x + y <= 35
x >= 0
y >= 0

using the desmos.com calculator, you graph the opposite of these constraints.

this means you will graph:

15x + 20y >= 600
x + y >= 35
x <= 0
y <= 0

the area of the graph that is NOT shaded is your region of feasibility.

the corner points of this region will be where your maximum revenue will be located.

your graph will look something like this:

$$$

the revenue at each corner point is calculated as shown below:

at (0,30), 2x + 2.5y becomes 2*0 + 30*2.5 = 75
at (20,15), 2x + 2.5y becomes 2*20 + 2.5*15 = 77.5
at (35,0), 2x + 2y becomes 2*35 + 0*2.5 = 70
at (0,0), 2x + 2y equals 0

she makes her most revenue when she sells 20 birthday cards and 15 holiday cards for revenue of 77.5 dollars.

her constraints at this corner point have to be satisfied as well.

15x + 20y <= 600 becomes 15*20 + 20*15 <= 600 which becomes 600 <= 600 which is true.

x + y <= 35 becomes 20 + 15 <= 35 which becomes 35 <= 35 which is true.

both x and y are greater than or equal to 0.

all the constraints are satisfied, so the answer is confirmed to provide the maximum revenue and satisfy all constraints at the corner point of (20,15).

she gets maximum revenue when she sells 20 birthday cards and 15 holiday cards.

the desmos.com calculator can be found at https://www.desmos.com/calculator

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