Chapter 7, Problem 17PS

(a)

To determine

The system of linear inequalities for the recommended cholesterol levels.

Answer to Problem 17PS

The system of linear inequalities for the recommended cholesterol levels are

  $\begin{array}{l}x+y\le 200\\ x\ge 60\\ 0\le y\le 130\end{array}$

Explanation of Solution

Given:

LDL level should be less than 130milligrams per deciliter and HDL level should be at least 60 milligrams per deciliter and total cholesterol 200 milligrams per deciliter. The system of inequalities cans DE:

Calculation:

If X represents HDL cholesterol and y represents LDL cholesterol, to find the system of linear inequalities are calculated below,

LDL level should be less than 130milligrams per deciliter and HDL level should be at least 60 milligrams per deciliter and total cholesterol 200 milligrams per deciliter. The system of inequalities cans DE:

  $\begin{array}{l}x+y\le 200\\ x\ge 60\\ 0\le y\le 130\end{array}$

Conclusion:

It depends upon the cholesterol levels and hence the equations are

  $x+y\le 200;x\ge 60;0\le y\le 130$

(b)

To determine

The graph of the system of inequalities.

Answer to Problem 17PS

The system of linear inequalities for the recommended cholesterol levels are

  $\begin{array}{l}x+y\le 200\\ x\ge 60\\ 0\le y\le 130\end{array}$

Explanation of Solution

Given:

LDL level should be less than 130milligrams per deciliter and HDL level should be at least 60 milligrams per deciliter and total cholesterol 200 milligrams per deciliter. The system of inequalities cans DE:

Calculation:

The system of linear inequalities for the recommended cholesterol levels are

  $\begin{array}{l}x+y\le 200\\ x\ge 60\\ 0\le y\le 130\end{array}$

To sketch the graph and label the vertices,

  

The graph has the vertices A(60,0); B(60,130); C(70,130); D(200,0).

Conclusion:

The graph has the vertices A(60,0); B(60,130); C(70,130); D(200,0).

(c)

To determine

The cholesterol levels are within the recommendations.

Answer to Problem 17PS

No, the given level is not well within the limit since the total cholesterol is greater than 200 milligram per deciliter.

Explanation of Solution

Given:

LDL level should be less than 130milligrams per deciliter and HDL level should be at least 60 milligrams per deciliter and total cholesterol 200 milligrams per deciliter. The system of inequalities cans DE:

Calculation:

To check the following levels come under the recommendation limits or not,

LDL-120 milligrams per deciliter

HDL-90 milligrams per deciliter

Total-210 milligrams per deciliter

No, the given level is not well within the limit since the total cholesterol is greater than 200 milligram per deciliter.

Conclusion:

No, the given level is not well within the limit since the total cholesterol is greater than 200 milligram per deciliter.

(d)

To determine

The example of the cholesterol levels if LDL/VLDL cholesterol level is too high but the HDL cholesterol level is acceptable.

Answer to Problem 17PS

The LDL level is so high, where VLDL and HDL level are in acceptable limits.

Explanation of Solution

Given:

LDL level should be less than 130milligrams per deciliter and HDL level should be at least 60 milligrams per deciliter and total cholesterol 200 milligrams per deciliter. The system of inequalities cans DE:

Calculation:

To find the cholesterol level where LDL level is too high and HDL and total cholesterol are in acceptable limits.

In order to make LDL in high level, make HDL level with some low acceptable level. Consider the HDL level as 60; the LDL level can be 140 milligram per deciliter, which is too high and total cholesterol is 200 milligram per deciliter. Another one can be HDL-135 milligram per deciliter and LDL-65 milligram per deciliter and total cholesterol is 200 milligram per deciliter.

Conclusion:

HDL-135 milligram per deciliter and LDL-65 milligram per deciliter and total cholesterol is 200 milligram per deciliter.

(e)

To determine

The point in the solution region that meets the recommendation.

Answer to Problem 17PS

The point $\left(75,105\right)$ is one such point which gives the ratio less than 5.

Explanation of Solution

Given:

LDL level should be less than 130milligrams per deciliter and HDL level should be at least 60 milligrams per deciliter and total cholesterol 200 milligrams per deciliter. The system of inequalities cans DE:

Calculation:

To find LDL and HDL level in such a way that the ratio or total cholesterol to HDL is less than 5. Consider some arbitrary point in the feasible solution set from the graph.

The point $\left(75,105\right)$ lies well within the solution region and $\frac{180}{75}=2.4<5$ .The HDL should be at least 60 and the feasible region starts from the X co- ordinate 60.

That is the ratio is well lesser than 5 at the lowest possible HDL level also. So all the points inside the region would be having the ratio less than 5 only

The point $\left(75,105\right)$ is one such point which gives the ratio less than 5.

Conclusion:

The point $\left(75,105\right)$ is one such point which gives the ratio less than 5.