1. Marissa is a photographer. She sells framed photographs for $100 each and greeting cards for $5 each. The materials for each framed photograph cost $30, and the materials for each greeting card cost $2. Marissa can sell up to 8 framed photographs and 40 greeting cards each week, but this week, she only has $200 to spend on materials. Marissa hopes to earn a profit of at lest $400 this week after paying for her materials.
Let x = the number of framed photographs.
Let y = the number of greeting cards.
Two of the inequalities that model this situation are x ≤ 8 and y ≤ 40 .
A. Write two more inequalities to complete the system of inequalities modeling this situation.
B. Explain your inequalities and explain why x ≤ 8 and y ≤ 40 are also inequalities for this system.
C. Graph the solution set to your system of inequalities on a coordinate plane. Shade the area that represents the solution set.
D. Marissa plans to make and sell 5 framed photographs and 25 greeting cards. Is that a solution to the system of inequalities? If so, is it the solution that will produce the most profit? Explain your answers.
no ideas on any of those questions?
A. To complete the system of inequalities, we need to add two more inequalities that represent the materials cost and the profit requirement.
1. Materials Cost Inequality:
The cost of materials for each framed photograph is $30, and the cost for each greeting card is $2. Since Marissa only has $200 to spend on materials, we can write the following inequality:
30x + 2y ≤ 200
2. Profit Requirement Inequality:
Marissa hopes to earn a profit of at least $400 after paying for her materials. The profit earned from selling each framed photograph is $100 - $30 (for materials) = $70. The profit from selling each greeting card is $5 - $2 (for materials) = $3. Therefore, the profit made from selling x framed photographs and y greeting cards can be expressed as:
70x + 3y ≥ 400
B. Inequality Explanation:
- x ≤ 8: This inequality represents the maximum number of framed photographs Marissa can sell, which is 8.
- y ≤ 40: This inequality represents the maximum number of greeting cards Marissa can sell, which is 40.
The reason these inequalities are included in the system is to set the upper limit on the number of framed photographs and greeting cards Marissa can sell within a week.
C. Graphing:
To graph the solution set, we can plot the inequalities on a coordinate plane. The x-axis can represent the number of framed photographs (x), and the y-axis can represent the number of greeting cards (y).
First, let's graph x ≤ 8 and y ≤ 40. To graph x ≤ 8, draw a vertical line at x = 8. To graph y ≤ 40, draw a horizontal line at y = 40. Shade the area below these lines to represent the solution set.
Next, let's graph 30x + 2y ≤ 200 and 70x + 3y ≥ 400. To do this, we need to plot the lines 30x + 2y = 200 and 70x + 3y = 400.
D. To check if Marissa's plan to make and sell 5 framed photographs and 25 greeting cards is a solution to the system of inequalities, we substitute x = 5 and y = 25 into the inequalities and check if they are satisfied.
For the inequalities, we have:
30x + 2y ≤ 200
30(5) + 2(25) ≤ 200
150 + 50 ≤ 200
200 ≤ 200
70x + 3y ≥ 400
70(5) + 3(25) ≥ 400
350 + 75 ≥ 400
425 ≥ 400
Both inequalities are satisfied, indicating that making and selling 5 framed photographs and 25 greeting cards is a valid solution to the system of inequalities.
To determine if it will produce the most profit, we can calculate the profit earned from selling 5 framed photographs and 25 greeting cards.
Profit = (Profit per framed photograph x Number of framed photographs) + (Profit per greeting card x Number of greeting cards)
Profit = (70 x 5) + (3 x 25)
Profit = 350 + 75
Profit = 425
The profit earned from selling 5 framed photographs and 25 greeting cards is $425, which is higher than the profit requirement of $400. Therefore, it is indeed the solution that will produce the most profit.