solve the word problem and graph it

Question

Lisa does a weekly exercise program consisting of cardiovascular work and weight training. Each week, she exercises for at least 12 hours. She spends at most 5 hours on weight training. She spends at most10 hours doing cardiovascular work. Let x denote the time (in hours) that Lisa spends doing cardiovascular work. Let y denote the time (in hours) that she spends on weight training. Shade the region corresponding to all values of x and y that satisfy these requirements.

Answer 1

Hi there! Great question :)

This problem already has a great start on it because they've already defined the variable for us. "X" will be the number of hours she spends on cardio and "y" will be the hours she spends on weight training. We will be using a system of equations to solve this problem.

Our first equation will come from the second sentence of the paragraph which says she spends at least 12 hours exercising. That means when we add up the time spent on cardio (x) and the time spent on weight training (y) our answer will be equal to or great than 12. Turning this into an equation we find:

x + y ≥ 12

Our second and third equations come from the statement saying that she spends at most 5 hours on weight training (y) and at most 10 hours on cardio (x). That means that our y must be less than or equal to 5 and our x must be less than or equal to 10:

y ≤ 5 x ≤ 10

We now have our three equations and can graph them. Let's take it one equation at a time.

To graph x + y ≥ 12, we can find the x and y-intercepts and then graph them. To find the x-intercept, we divide the 12 by the x coefficient which is one. Doing so gives us an x-intercept of 12. So our first point will be on the x-axis (the horizontal axis) at the point 12. To find the y-intercept we do the same thing; divide 12 by the y coefficient which is also one and give us another answer of 12. So this time we will put a point on the y axis (the vertical axis) at 12. Now since the inequality symbol is or equal to, that tells us our line should be solid on our graph. Connect the two intercept with a straight line. Now for shading. Since our x and y is greater than 12, we will shade ABOVE our line. You are now done with the graphing of this equation.

The last two equations will be simple to graph. First let's do y≤5. All you're going to do is draw a solid, horizontal line that crosses the point 5 on the y-axis (again, the vertical axis). Since the y is less than 5, you will shade BELOW this line.

Finally we can graph x ≤ 10. This time, we will be drawing a vertical line through the point 10 on the x-axis (the horizontal axis. Again, x is less than 10, so this time we will draw to the LEFT of the line we just graphed.

Now the region that satisfies the requirements of this problem will be the area that is shaded by all three equations.

I hope this explanation was easy to follow. If you have any questions please don't hesitate to reach out! Learn on! :)