What are the restrictions for graphing Ramesh's exercise plan?

[might hire]

Okay, here is the problem, ; Remesh likes to run outdoors and ride his bicycle at the veledrome. He burns about 400 calories/h running, and 300/h riding his bike. It costs $6/h to ride in the veledrome. Ramesh hopes to develope a weekly exersise program that will burn 4800 calories, cost no more than $24, and require a maximum of 8 hours... Now I need the restrictions to graph this problem, (x=running/h; y=bike/h) I so far have; x (> or equal to) 0, y (> or equal to) 0, x+y (< or equal to) 8, and y (< or equal to) 4. I think I'm missing one, and think it might have to do with the calories, so is 400x+300y (> or equal to) 4800. But would it be applicable to the hour restriction? Thanks in Advance!

 

[verty]

Intuitively, the most calories will be burnt by running. 8 hours of running will burn 3200 calories, so it is impossible to get 4800 from 8 hours of exercise.

When you plot the inequalities, the solution lies in the intersection of the areas. When you plot the fifth inequality (400x + 300y >= 4800) you'll find the areas do not intersect, so there is no solution.

 

[tacman]

Your restrictions so far are correct. In order to incorporate the calorie restriction, you can add another inequality: 400x + 300y (greater than or equal to) 4800. This represents the minimum number of calories that need to be burned in order to meet Ramesh's goal of 4800 calories per week.

This restriction is applicable to the hour restriction because it ensures that Ramesh is not exceeding the maximum of 8 hours per week while still meeting his calorie goal.

So your final set of restrictions would be: x (greater than or equal to) 0, y (greater than or equal to) 0, x+y (less than or equal to) 8, y (less than or equal to) 4, 400x + 300y (greater than or equal to) 4800.

Graphing these restrictions will give you a feasible region where the x and y values represent the number of hours spent running and biking, respectively. Ramesh's optimal exercise plan would be to choose a point within this feasible region that satisfies all the restrictions and minimizes his cost.