Activity 1: Graphing
To fill an order for Sizzlin' Sauce sauces, you bought 1050 green peppers and 1200 hot chili peppers.
• Write and graph a system of inequalities to represent how many pints of each kind of sauce you can make. Use the recipes below.
• Select one solution of the system and determine how many peppers you will have left over.
Sizzlin' Sauces Recipes
Scorchin' Hot Sauce Ingredients Red Hot Sauce Ingredients
Yield: 1 pint
1 pint tomato sauce with onions
4 green peppers, diced
8 hot chili peppers, seeded and diced Yield: 1 pint
1 pint tomato sauce with onions
5 green peppers, diced
4 hot chili peppers, seeded and diced
If there are p pints of sauce #1 and q pints of sauce #2,
4p+5q <= 1050
8p+4q <= 1200
Now just pick a solution and proceed.
To graph a system of inequalities to represent how many pints of each kind of sauce you can make, we first need to define our variables. Let's say:
- x represents the number of pints of Scorchin' Hot Sauce.
- y represents the number of pints of Red Hot Sauce.
Now, let's set up the inequalities based on the given recipes.
For Scorchin' Hot Sauce:
- The number of green peppers needed is 4x.
- The number of hot chili peppers needed is 8x.
The total number of green peppers used for Scorchin' Hot Sauce will be:
4x
The total number of hot chili peppers used for Scorchin' Hot Sauce will be:
8x
Therefore, the inequality for the green peppers used is:
4x ≤ 1050
And the inequality for the hot chili peppers used is:
8x ≤ 1200
For Red Hot Sauce:
- The number of green peppers needed is 5y.
- The number of hot chili peppers needed is 4y.
The total number of green peppers used for Red Hot Sauce will be:
5y
The total number of hot chili peppers used for Red Hot Sauce will be:
4y
Therefore, the inequality for the green peppers used is:
5y ≤ 1050
And the inequality for the hot chili peppers used is:
4y ≤ 1200
Now, let's graph the inequalities on a coordinate plane. Plot the points where each inequality intersects the x and y axes, and shade the areas that satisfy the inequalities.
Once you have graphed the system of inequalities, you can find a solution by selecting a point within the shaded region. For example, let's say we choose x = 100 and y = 150 as a solution point.
To determine how many peppers you will have left over, substitute these values into the equations for each recipe and calculate the remaining peppers:
For Scorchin' Hot Sauce:
- The number of green peppers used is 4 * 100 = 400.
- The number of hot chili peppers used is 8 * 100 = 800.
For Red Hot Sauce:
- The number of green peppers used is 5 * 150 = 750.
- The number of hot chili peppers used is 4 * 150 = 600.
Now, calculate the remaining peppers:
- Number of green peppers left = Total green peppers - Green peppers used
= 1050 - 400
= 650
- Number of hot chili peppers left = Total hot chili peppers - Hot chili peppers used
= 1200 - 800
= 400
So, if you make 100 pints of Scorchin' Hot Sauce and 150 pints of Red Hot Sauce, you will have 650 green peppers and 400 hot chili peppers left over.