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Can you help with Algebra 1 system of inequalities (money for college project) *urgent*?
Word problem: "john is going to be a senior in high school and is planning to help pay for his way through college. he decided to get two jobs over the summer so that he could make as much money as possible. johns main job involves working as a lifegaurd making $15 per hour. when he has free time on weekends, he mows lawns charging $10 per hour. John wants to save a minimum of $200 per week. unforunately, he is taking a summer school class every morning from 8 am to 12pm, so the most he can work each week is 20 hours.
the system of linear equlities would be y is less than or equal to x+20 and y is greater than or equal to -15/10 x+20
I already graphed it, but i dont know WHY IT IS IN QUADRENT ONE.
IS IT POSSIBLE FOR JOHN TO REACH HIS GOAL OF MAKING AT LEAST $200 PER WEEK? EXPLAIN.
WHAT ARE THREE POSSIBLE SOLUTIONS? WHAT DOES EACH SOULTION MEAN IN TERMS OF HOURS WORKED AND MONEY MADE.
WHAT IS THE MOST MONEY THAT JOHN CAN MAKE WORKING NO MORE THAN 20 HRS A WEEK? HOW MANY HOURS AT EACH JOB WOULD HELP HIM REACH THAT AMOUNT?
WHAT SOLUTION WOULD ALLOW HIM TO MAKE EXACTLY $200 WORKING AS FEW HOURS AS POSSIBLE? HOW MANY HOURS OF EACH JOB WOULD HE HAVE TO WORK? HOW MANY TOTAL HOURS?
if you can answer any/ all of these questions, i will be xtremely greatful. i need the answers as soon as possible. thank you so much for your help! please do your best and thank you.
1 Answer
- I know some mathLv 41 decade agoFavorite Answer
Could your first inequality be wrong?
x + y <= 20
and
15x + 10y >= 200, so we need to graph
y <= -x + 20 (you had x + 20 for this one)
and
y >= (-3/2)x + 20.
On top of that, as a common sense, it is plausible
to assume that x >= 0 and y >= 0 as Joh can't work negative number
of hours. That explains why we are also restricted to the quadrant I.
If you graph the region, we get a shaded triangle. Any point selected
in this triangle (including points on the boundary) represent a feasible
solution the problem. For your information the vertices of the triangle
are (0, 20), (40/3, 0), and (20, 0). According to the Fundamental Theorem
of Linear Programming (or the Vertex Theorem), we get the extreme
values of the goal function (F(x, y) = 15x + 10y in this problem) at
the vertices of the feasible region (the shaded triangle including the
boundary in this problem).
At (0, 20), F(0, 20) = 15(0) + 10(20) = 200.
At (40/3, 0), F(40/3, 0) = 15(40/3) + 10(0) = 200.
At (20, 0), F(20, 0) = 15(20) + 10(0) = 300.
Hence, John can earn up to $300 if he works 20 hours as a life-guard
and forget about mowing.
