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: -
Total Hours Inequality:
The total hours worked (both babysitting and mowing) cannot exceed 50 hours: -
Non-negativity Constraints:
Both and must be non-negative (she can't work negative hours):
To find combinations of
We can simplify it by dividing the entire inequality by 3:
Now we need to consider the two inequalities:
The goal is to minimize
To find the intersection points of the inequalities, we will find the lines
Finding Intersection Points:
-
From the earnings inequality: We can express
as: -
From the total hours inequality: We can express
as:
Setting the Equations Equal:
Now, we can set these equal to each other to find the intersection:
Multiplying through by 5 to eliminate the fraction:
Rearranging gives:
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
into : -
Substituting
into : -
Now check
:
Substituting into
Let's check the two valid points to find minimum
- For
: - For
Thus the feasible region points gives us minimum:
Final Validation:
Checking at
- Hours:
hours (minimum selection)
Hence the minimum number of total hours she can work to earn at least $300 while not exceeding 50 hours is \( 20 hours.\)