Math  /  Algebra

QuestionCool Down Lin's job pays $8.25\$ 8.25 an hour plus $10\$ 10 of transportation allowance each week. She has to work at least 5 hou week to keep the job, and can earn up to $175\$ 175 per week (including the allowance).
1. Represent this situation mathematically. If you use variables, specify what each one means.
2. How many hours per week can Lin work? Explain or show your reasoning.

Studdy Solution

STEP 1

What is this asking? How many hours can Lin work each week, if she earns $8.25\$8.25 an hour plus $10\$10 weekly, must work at least 5 hours, and can earn at most $175\$175 per week? Watch out! Don't forget about the minimum and maximum number of hours Lin can work!

STEP 2

1. Define the variables and function
2. Set up the inequality
3. Solve for the number of hours

STEP 3

Let's **define** \(h\) as the **number of hours** Lin works per week.
This is what we want to find out!

STEP 4

Lin's **total earnings** per week can be represented by the **function**: E(h)=8.25h+10 E(h) = 8.25 \cdot h + 10 where E(h)E(h) is the total earnings and hh is the number of hours worked.
We **multiply** the **hourly rate** \(8.25\) by the **number of hours** \(h\) and **add** the $10\$10 **transportation allowance**.

STEP 5

Lin must earn no more than $175\$175 per week.
So, her total earnings, E(h)E(h), must be **less than or equal to** $175\$175.
We can write this as: E(h)175 E(h) \le 175

STEP 6

We also know Lin must work *at least* **5 hours** a week.
So, hh must be **greater than or equal to** \(5\).
We can write this as: h5 h \ge 5

STEP 7

Substituting our expression for E(h)E(h) from the earnings function, we get: 8.25h+10175 8.25 \cdot h + 10 \le 175

STEP 8

Let's **solve** for \(h\)!
First, we **subtract** \(10\) from both sides of the inequality: 8.25h+101017510 8.25 \cdot h + 10 - 10 \le 175 - 10 8.25h165 8.25 \cdot h \le 165

STEP 9

Next, we **divide** both sides by \(8.25\) to **isolate** \(h\): 8.25h8.251658.25 \frac{8.25 \cdot h}{8.25} \le \frac{165}{8.25} h20 h \le 20

STEP 10

So, Lin can work at most **20 hours** per week.
Remembering that she must also work at least **5 hours**, we can write the final solution as: 5h20 5 \le h \le 20

STEP 11

Lin can work between **5 and 20 hours** per week.

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