Lynsay plans to earn at least $300 babysitting and mowing lawns this summer. She will charge $12 per hour for babysitting, and $15 per hour for mowing lawns. She would like to work a maximum of 50 hours this summer.

Question

She wrote the following system of inequalities to determine combinations of jobs she can do. Let x = number of hours spent babysitting, and y = number of hours spent mowing lawns.

What is the smallest number of total hours she can work in order to earn at least $300 without working more than 50 hours?

GPT-4o mini · Answer

To determine the smallest number of total hours Lynsay can work while earning at least $300 without exceeding 50 hours, we can set up the following inequalities based on her situation:

Earnings Inequality:
The earnings from babysitting at $12 per hour and mowing lawns at $15 per hour must be at least $300: $12 x + 15 y \geq 300$
Total Hours Inequality:
The total hours worked (both babysitting and mowing) cannot exceed 50 hours: $x + y \leq 50$
Non-negativity Constraints:
Both $x$ and $y$ must be non-negative (she can't work negative hours): $x \geq 0, y \geq 0$

To find combinations of $x$ and $y$ that satisfy the equations, we will first rewrite the earnings inequality:

$12 x + 15 y \geq 300$

We can simplify it by dividing the entire inequality by 3:

$4 x + 5 y \geq 100$

Now we need to consider the two inequalities:

$4 x + 5 y \geq 100$
$x + y \leq 50$

The goal is to minimize $x + y$ , which represents the total hours worked.

To find the intersection points of the inequalities, we will find the lines $4 x + 5 y = 100$ and $x + y = 50$ .

Finding Intersection Points:

From the earnings inequality: We can express $y$ as: $y = \frac{100 - 4 x}{5}$
From the total hours inequality: We can express $y$ as: $y = 50 - x$

Setting the Equations Equal:

Now, we can set these equal to each other to find the intersection: $\frac{100 - 4 x}{5} = 50 - x$

Multiplying through by 5 to eliminate the fraction: $100 - 4 x = 250 - 5 x$

Rearranging gives: $5 x - 4 x = 250 - 100$ $x = 150$

However, this value does not make sense in our hour bounds. We might have miscalculated there. Let's solve stepwise for points of intersection:

To find the feasible solutions, we can check values:

Substituting $y = 0$ into $4 x + 5 y \geq 100$ : $4 x \geq 100 ⟹ x \geq 25, \Rightarrow (25, 0)$
Substituting $x = 0$ into $4 x + 5 y \geq 100$ : $5 y \geq 100 ⟹ y \geq 20, \Rightarrow (0, 20)$
Now check $x + y = 50$ : $x + y = 50, \Rightarrow y = 50 - x$

Substituting into $4 x + 5 (50 - x) \geq 100$ : $4 x + 250 - 5 x \geq 100$ $- x + 250 \geq 100 ⟹ - x \geq - 150 ⟹ x \leq 150$

Let's check the two valid points to find minimum $x + y$ : So we checkon $(25, 0)$ and $(0, 20)$ :

For $x + y$ : $25 + 0 = 25$
For $0 + 20 = 20$

Thus the feasible region points gives us minimum:

Final Validation:

Checking at $x = 0 / y = 20$

Hours: $20$ hours (minimum selection)

Hence the minimum number of total hours she can work to earn at least $300 while not exceeding 50 hours is $ 20 hours.$