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Zorluk: OrtaSystems of Linear Inequalities in Two Variables

A student works two part-time jobs: tutoring, which pays 2020 dollars per hour, and working at a bookstore, which pays 1212 dollars per hour. The student can work at most 1515 hours per week and wants to earn at least 220220 dollars per week. If the student works a whole number of hours at each job, what is the minimum number of hours the student must tutor to meet these requirements?

Cevap: 5 hours

Cevap

5
The system of inequalities modeling the scenario consists of x+y≤15x + y \leq 15 (representing the limit on total hours) and 20x+12y≥22020x + 12y \geq 220 (representing the earning target), where xx is the hours of tutoring and yy is the hours at the bookstore. To find the minimum hours of tutoring, we assume the maximum bookstore hours y=15−xy = 15 - x and substitute it into the earnings equation: 20x+12(15−x)≥22020x + 12(15 - x) \geq 220. Simplifying yields 8x+180≥2208x + 180 \geq 220, which reduces to 8x≥408x \geq 40, or x≥5x \geq 5. The minimum integer value that satisfies this condition is 5.

Adım Adım Çözüm

1
Define variables and set up the constraint for total hours worked.
x+y≤15x + y \leq 15
Let xx be the number of hours tutoring and yy be the number of hours working at the bookstore. The total hours cannot exceed 15.
2
Set up the constraint for the minimum weekly earnings.
20x+12y≥22020x + 12y \geq 220
Tutoring pays 2020 dollars per hour and the bookstore pays 1212 dollars per hour, and the total earnings must be at least 220220 dollars.
3
Substitute the maximum value of yy in terms of xx into the earnings inequality.
20x+12(15−x)≥22020x + 12(15 - x) \geq 220
To minimize xx, we must maximize yy. From x+y≤15x + y \leq 15, the maximum value of yy is 15−x15 - x.
4
Solve the inequality for xx.
x≥5x \geq 5
Distribute and simplify: 20x+180−12x≥220⇒8x+180≥220⇒8x≥40⇒x≥520x + 180 - 12x \geq 220 \Rightarrow 8x + 180 \geq 220 \Rightarrow 8x \geq 40 \Rightarrow x \geq 5.

Anahtar Kavram

Solving systems of linear inequalities to find optimal boundary values in context.
Tahmini Süre:1m 30s
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Systems of Linear Inequalities in Two Variables

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