Select The Correct Answer.Zahra Is Knitting Items To Sell At A Craft Fair. She Has A Total Of 2,640 Yards Of Yarn. A Scarf Uses 200 Yards Of Yarn, And A Hat Uses 150 Yards. She Wants To Knit A Minimum Of 15 Items.This System Of Inequalities Represents
**Select the Correct Answer: Zahra's Yarn Conundrum**
Zahra is an ambitious craftswoman who wants to make the most of her yarn stash at the upcoming craft fair. With 2,640 yards of yarn at her disposal, she needs to decide how many scarves and hats to knit in order to meet her minimum sales target of 15 items. In this article, we will explore the system of inequalities that represents Zahra's yarn conundrum and help her make an informed decision.
Zahra has two types of items to knit: scarves and hats. Each scarf requires 200 yards of yarn, while each hat requires 150 yards. She wants to knit a minimum of 15 items, but she also needs to consider the total amount of yarn she has available. Let's denote the number of scarves as S and the number of hats as H.
The system of inequalities that represents Zahra's yarn conundrum can be written as:
- 200S + 150H ≥ 15 (minimum number of items)
- 200S + 150H ≤ 2640 (total amount of yarn available)
Let's break down each inequality and understand what it represents:
- 200S + 150H ≥ 15: This inequality states that the total number of items (scarves and hats) must be at least 15. Since each scarf requires 200 yards of yarn and each hat requires 150 yards, the total amount of yarn used must be at least 15 times the minimum amount required for each item.
- 200S + 150H ≤ 2640: This inequality states that the total amount of yarn used must not exceed 2640 yards. Since Zahra has 2640 yards of yarn available, this inequality ensures that she doesn't run out of yarn.
To solve the system of inequalities, we can use a graphical approach or algebraic methods. Let's use a graphical approach to visualize the solution.
Graphical Approach
We can plot the two inequalities on a coordinate plane, with S on the x-axis and H on the y-axis. The first inequality (200S + 150H ≥ 15) represents a line with a slope of -200/150 = -4/3 and a y-intercept of 15/150 = 1/10. The second inequality (200S + 150H ≤ 2640) represents a line with a slope of -200/150 = -4/3 and a y-intercept of 2640/150 = 17.6.
Algebraic Approach
We can also solve the system of inequalities using algebraic methods. Let's rewrite the first inequality as:
200S + 150H = 15 + k
where k is a non-negative constant. We can then rewrite the second inequality as:
200S + 150H = 2640 - k
Finding the Solution
To find the solution, we need to find the values of S and H that satisfy both inequalities. We can do this by finding the intersection of the two lines.
Intersection of the Lines
The intersection of the two lines can be found by setting the two equations equal to each other:
200S + 150H = 15 + k 200S + 150H = 2640 - k
Solving for S and H, we get:
S = 12 H = 6
Zahra's yarn conundrum can be represented by a system of inequalities. By solving the system of inequalities, we can find the values of S and H that satisfy both inequalities. In this case, Zahra should knit 12 scarves and 6 hats to meet her minimum sales target of 15 items.
Q: What is the minimum number of items Zahra needs to knit? A: Zahra needs to knit a minimum of 15 items.
Q: How many yards of yarn does each scarf require? A: Each scarf requires 200 yards of yarn.
Q: How many yards of yarn does each hat require? A: Each hat requires 150 yards of yarn.
Q: What is the total amount of yarn available? A: Zahra has 2640 yards of yarn available.
Q: How can Zahra solve the system of inequalities? A: Zahra can solve the system of inequalities using a graphical approach or algebraic methods.
Q: What is the solution to the system of inequalities? A: The solution to the system of inequalities is S = 12 and H = 6.
Q: What should Zahra do to meet her minimum sales target? A: Zahra should knit 12 scarves and 6 hats to meet her minimum sales target of 15 items.