Zach is planning to invest up to $50,000 in corporate and municipal bonds. The lest he will invest in corporate bonds is $6000 and he does not want to invest more than $27,000 in corporate bonds. He also does not want to invest more than $34,650 in municipal bonds. The interest is 8.5% on corporate bonds and 6.8% on municipal bonds. This is simple interest for one year. What is the maximum income?
Graph
solve using linear programming.
i already got that the equation is I=.085*27000+.068*23000, equals $3859, I just dont know how to graph these
maximize
.085 x + .068 y
x+y <= 50,000
x>=6,000
x<=27,000
y<=34,650
x+y>=1 phony to fill fields
I get $3859 at (27000, 23000)
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How did you get 23000
Well, Zach seems to have a lot on his plate with all these investment decisions. But don't worry, I am here to help, with a touch of humor, of course!
Let's break it down. We have two variables: the amount invested in corporate bonds, c, and the amount invested in municipal bonds, m. Our objective is to maximize the income, I. The interest rates are given as 8.5% for corporate bonds and 6.8% for municipal bonds.
Based on the given restrictions, we have the following equations:
0 ≤ c ≤ 27000
0 ≤ m ≤ 34650
c + m ≤ 50000
Now, let's tackle this graphically. Imagine a coordinate plane, where c represents the x-axis and m represents the y-axis. We can plot the feasible region by graphing the inequalities.
Since graphing on text-based platforms is quite challenging, let me describe the process for you. Draw a rectangle that spans from (0, 0) to (27000, 34650). Then, draw a line with a slope of -1 passing through (27000, 0), and continue it until it intersects the rectangle. This represents the equation c + m ≤ 50000. Shade the region below this line within the rectangle.
Next, draw a horizontal line at y = 34650. Shade the region to the left of this line within the rectangle to represent the inequality m ≤ 34650.
Finally, draw a vertical line at x = 27000. Shade the region below this line within the rectangle to represent the inequality c ≤ 27000.
The feasible region is the overlapping shaded area within the rectangle.
Now, to determine the maximum income, we can calculate the values of I at each vertex of the feasible region and choose the highest one.
Once again, the equation for income is I = 0.085c + 0.068m. Substitute the values for c and m at each vertex into this equation, and find the maximum value.
And voila! You'll have your maximum income within the investment constraints.
Remember, investing can be quite a juggling act, so make sure to do your calculations carefully. Good luck, and may your investments bring you more laughs than tears!
To graph the given constraint equations, you need to plot the inequality on a coordinate plane. The X-axis represents the investment in corporate bonds, and the Y-axis represents the investment in municipal bonds.
Let's label the axis as follows:
X-axis: Corporate Bonds Investment (in thousands of dollars)
Y-axis: Municipal Bonds Investment (in thousands of dollars)
Now let's graph the constraints:
1. The least investment in corporate bonds is $6,000, which means X ≥ 6.
Plot a vertical line passing through X = 6.
2. The maximum investment in corporate bonds is $27,000, which means X ≤ 27.
Plot a vertical line passing through X = 27.
3. The maximum investment in municipal bonds is $34,650, which means Y ≤ 34.65.
Plot a horizontal line passing through Y = 34.65.
The feasible region, or the region where all the constraints are satisfied, is the area enclosed by these three lines.
Next, we'll calculate the maximum income by using the objective function. In this case, the objective function is:
I = 0.085X + 0.068Y
To find the maximum income, you'll need to evaluate the objective function at each corner point of the feasible region and pick the highest value.
I hope this helps!
To graph the given problem using linear programming, we will need to set up a feasible region and then find the corner points of that region to determine the maximum income.
Here are the steps to graph the problem:
Step 1: Define the variables:
Let's define two variables: x for the amount invested in corporate bonds and y for the amount invested in municipal bonds.
Step 2: Establish the constraints:
Based on the given information, we have the following constraints:
- Corporate bond investment: 6000 ≤ x ≤ 27000
- Municipal bond investment: y ≤ 34650
- Total investment: x + y ≤ 50000
Step 3: Plot the constraints:
Plot each constraint on a graph by converting them into equations and then graphing the corresponding lines or curves.
- Corporate bond investment: Plot the line x = 6000 and x = 27000.
- Municipal bond investment: Plot the line y = 34650.
- Total investment: Plot the line x + y = 50000.
Step 4: Shade the feasible region:
Since Zach wants to invest up to $50,000, the feasible region will be the region that satisfies all the constraints. Shade the region that satisfies all the constraints.
Step 5: Find corner points:
Locate the corner points of the shaded region. These are the intersections of the lines or curves that represent the constraints.
Step 6: Evaluate the objective function:
Calculate the value of the objective function (maximum income) at each corner point. The corner point with the highest value of the objective function will be the maximum income.
Step 7: Determine the maximum income:
Evaluate the objective function (maximum income) at each corner point and select the point with the highest value. In this case, you can plug the values of x and y from the corner points into the objective function: I = 0.085x + 0.068y.
Choose the corner point that yields the highest value for I. That will be your answer for the maximum income.
Note: In your question, the equation you provided to calculate the maximum income seems incorrect. The correct equation is I = 0.085x + 0.068y.