Math Problem Statement
Bonar memiliki dua pekerjaan paruh waktu. Untuk mengantar barang, Bonar dibayar Rp15.000,00 per jam. Untuk pekerjaan mencuci piring di restoran, Bonar dibayar Rp9.000,00 per jam. Dia membutuhkan uang sebesar Rp120.000,00. Dia tidak dapat bekerja lebih dari 10 jam. Berapa jam dia harus bekerja untuk masing-masing pekerjaan?
Solution
Let's solve the problem in the image step by step.
Problem Summary:
Bonar has two part-time jobs:
- Job 1: Delivering goods, for which he is paid Rp15,000 per hour.
- Job 2: Washing dishes in a restaurant, for which he is paid Rp9,000 per hour.
He needs to earn a total of Rp120,000. However, Bonar cannot work more than 10 hours combined between both jobs. The goal is to determine how many hours he must work at each job to meet his financial target.
(a) Mathematical Model:
Let:
- = hours worked delivering goods.
- = hours worked washing dishes.
The total earnings from both jobs is:
Bonar's total working hours constraint is:
Therefore, the mathematical model is:
15,000x + 9,000y = 120,000 \\ x + y \leq 10 \end{cases}$$ --- ### (b) **Is the model linear?** Yes, this is a system of linear equations. Both the equations involve variables $$x$$ and $$y$$ in the first degree (no exponents), which defines a linear system. --- ### (c) **Graph of the system:** To plot this system, we would graph both the equations. The line $$15,000x + 9,000y = 120,000$$ represents the total earnings constraint, and the line $$x + y = 10$$ represents the total hours constraint. We can solve these equations graphically by finding the intercepts and plotting the lines. 1. For the earnings equation: - When $$x = 0$$: $$9,000y = 120,000 \implies y = \frac{120,000}{9,000} = 13.33$$ - When $$y = 0$$: $$15,000x = 120,000 \implies x = \frac{120,000}{15,000} = 8$$ 2. For the hours constraint $$x + y = 10$$: - When $$x = 0$$, $$y = 10$$ - When $$y = 0$$, $$x = 10$$ Now, these two lines can be plotted to find the feasible region. --- ### (d) **Finding the points of intersection:** To find where the lines intersect, solve the system algebraically: 1. From $$x + y = 10$$, we get $$y = 10 - x$$. 2. Substitute this into the earnings equation: $$15,000x + 9,000(10 - x) = 120,000$$ Simplifying: $$15,000x + 90,000 - 9,000x = 120,000$$ $$6,000x = 30,000$$ $$x = 5$$ 3. Substitute $$x = 5$$ into $$x + y = 10$$ to get $$y = 5$$. Thus, the point of intersection is $$(5, 5)$$, meaning Bonar must work **5 hours delivering goods** and **5 hours washing dishes** to meet his goal of earning Rp120,000 within the 10-hour limit. --- Would you like more detailed explanations on any part of the solution? Here are some related questions for practice: 1. What is the geometric interpretation of the system of equations in this problem? 2. How would the solution change if Bonar needed to earn Rp150,000 instead of Rp120,000? 3. If Bonar was only allowed to work a maximum of 8 hours, how would the equations change? 4. Can you verify the solution by substitution in the original system of equations? 5. What if Bonar earned Rp18,000 per hour for delivering goods, how would that affect the model? **Tip:** When solving word problems involving systems of equations, identify constraints and objectives carefully before setting up the mathematical model.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Equations
Systems of Equations
Graphing
Formulas
15,000x + 9,000y = 120,000
x + y ≤ 10
Theorems
Linear System Theorem
Suitable Grade Level
Grades 10-12