11. As part of a weight reduction program, a man designs a monthly exercise program consisting of bicycling, jogging, and swimming. He would like to exercise at most 38 hours, devote at most 3 hours to swimming, and jog for no more than the total number of hours bicycling and swimming. The calories burned per hour by bicycling, jogging, and swimming are 200, 622, and 258, respectively. How many hours should be allotted to each activity to maximize the number of calories burned? If he loses 1 pound of weight for each 3,500 calories burned, how many pounds will he lose each month exercising?
How many hours should be allotted to each activity to maximize the number of calories burned?
_ hours for swimming
_ hours for bicycling
_ hours for jogging
To maximize the number of calories burned, we need to form a linear programming problem.
Let's define the following variables:
x = number of hours for swimming
y = number of hours for bicycling
z = number of hours for jogging
We need to maximize the objective function: Calories burned = 258x + 200y + 622z
Subject to the following constraints:
x + y + z ≤ 38 (He can exercise at most 38 hours)
x ≤ 3 (He can devote at most 3 hours to swimming)
z ≤ y + x (He can jog for no more than the total hours of swimming and cycling)
Now, let's solve the linear programming problem to find the optimal solution.
Since this is a linear programming problem, we can use the simplex method to find the optimal solution. However, I'll solve it graphically for simplicity.
First, let's plot the feasible region based on the given constraints:
The equation x + y + z = 38 represents the line with the highest value of x, y, and z within the constraint x + y + z ≤ 38.
The equation x = 3 represents the vertical line where x is limited to 3 hours.
The equation z = y + x represents the line where the z value is equal to the sum of x and y.
After plotting these three lines, we can find the feasible region, which is the triangular area bounded by these lines.
Now, to maximize the calories burned, we need to find the vertex points of the feasible region and evaluate the objective function at each vertex.
The vertex points of the feasible region are (3, 0, 35), (3, 3, 32), and (0, 5, 33).
Now, let's calculate the calories burned at each vertex point:
For (3, 0, 35):
Calories burned = 258(3) + 200(0) + 622(35) = 21,690
For (3, 3, 32):
Calories burned = 258(3) + 200(3) + 622(32) = 22,532
For (0, 5, 33):
Calories burned = 258(0) + 200(5) + 622(33) = 25,910
Based on these calculations, we can see that the maximum number of calories burned is 25,910, which occurs when he allots:
0 hours for swimming
5 hours for bicycling
33 hours for jogging
Therefore, he should allocate 0 hours to swimming, 5 hours to bicycling, and 33 hours to jogging to maximize the number of calories burned.
Now let's calculate how many pounds he will lose each month by using the conversion rate of 1 pound for each 3,500 calories burned.
Calories burned each month = Calories burned per hour * Number of hours allocated
Calories burned each month = 622 * 5 + 258 * 33
Calories burned each month = 3,110 + 8,514
Calories burned each month = 11,624
Pounds lost each month = Calories burned each month / 3,500
Pounds lost each month = 11,624 / 3,500
Pounds lost each month ≈ 3.32 pounds
Therefore, he will lose approximately 3.32 pounds each month by exercising.
To determine how many hours should be allotted to each activity to maximize the number of calories burned, we can set up a linear programming problem. Let's represent the variables as follows:
x: hours for swimming
y: hours for bicycling
z: hours for jogging
The problem provides the following constraints:
1. The total exercise time should be at most 38 hours: x + y + z ≤ 38
2. The time allotted for swimming should be at most 3 hours: x ≤ 3
3. The jogging time should be less than or equal to the total hours of bicycling and swimming: z ≤ y + x
To maximize the number of calories burned, we need to maximize the objective function, which is the total calories burned.
Given that the calories burned per hour are:
Swimming: 258 calories/hour
Bicycling: 200 calories/hour
Jogging: 622 calories/hour
Therefore, the objective function is:
Objective function = 258x + 200y + 622z
This problem can be solved using a linear programming solver or graphical method to find the optimal solution. The optimal solution will provide the values of x, y, and z that maximize the objective function while satisfying all the constraints.
Hence, you can use a linear programming solver or graphing techniques to find the exact values of x, y, and z to maximize the number of calories burned.
Please note that I am unable to show the calculations or provide you with the exact values without knowing the specific values of x, y, and z. You can use a linear programming solver, such as the Simplex Method or graphical techniques, to find the optimal solution.
As part of a weight reduction program, a man designs a monthly exercise program consisting of bicycling, jogging, and swimming. He would like to exercise at most 38 hours, devote at most 8 hours to swimming, and jog for no more than the total number of hours bicycling and swimming. The calories burned per hour by bicycling, jogging, and swimming are 200, 637, and 273, respectively. How many hours should be allotted to each activity to maximize the number of calories burned? If he loses 1 pound of weight for each 3,500 calories burned, how many pounds will he lose each month exercising?
How many hours should be allotted to each activity to maximize the number of calories burned?
nothing hours for swimming
nothing hours for bicycling
nothing hours for jogging
How many pounds will he lose each month exercising?
nothing pounds (Round to the nearest pound as needed.)