Chapter 7, Problem 17PS

a.

To determine

Write a system of linear inequalities.

Answer to Problem 17PS

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

Explanation of Solution

Given information:

Cholesterol in human blood is necessary, but too much can lead to health problems. There are three main types of cholesterol: HDL (high-density lipoproteins), LDL (low-density lipoproteins), and VLDL (very low-density lipoproteins). HDL is considered “good” cholesterol; LDL and VLDL are considered “bad” cholesterol. A standard fasting cholesterol blood test measures total cholesterol, HDL cholesterol, and triglycerides. These numbers are used to estimate LDL and VLDL, which are difficult to measure directly. It is recommended that your combined LDL/VLDL cholesterol level be less than $130$ milligrams per decilitre, your HDL cholesterol level be at least $40$ milligrams per decilitre, and your total cholesterol level be no more than 200 milligrams per decilitre.

Write a system of linear inequalities for the recommended cholesterol levels. Let represent the HDL cholesterol level, and let represent the combined LDL VLDL cholesterol level.

Calculation:

If $x$ represents HDL cholesterol and $y$ represents LDL cholesterol, we need to find the system of inequalities.

LDL level should be less than $130$ milligrams per decilitre and HDL level should be at least $60$ milligrams per decilitre and total cholesterol $200$ milligrams per decilitre.

Hence ,the system of inequalities can be: $\boxed{\begin{array}{l}x+y\le 200\\ x\ge 60\\ 0\le y\le 130\end{array}}$

b.

To determine

Graph the system of inequalities.

Answer to Problem 17PS

  

Explanation of Solution

Given information:

Cholesterol in human blood is necessary, but too much can lead to health problems. There are three main types of cholesterol: HDL (high-density lipoproteins), LDL (low-density lipoproteins), and VLDL (very low-density lipoproteins). HDL is considered “good” cholesterol; LDL and VLDL are considered “bad” cholesterol. A standard fasting cholesterol blood test measures total cholesterol, HDL cholesterol, and triglycerides. These numbers are used to estimate LDL and VLDL, which are difficult to measure directly. It is recommended that your combined LDL/VLDL cholesterol level be less than $130$ milligrams per decilitre, your HDL cholesterol level be at least $40$ milligrams per decilitre, and your total cholesterol level be no more than 200 milligrams per decilitre.

Graph the system of inequalities from part (a).Label any vertices of the solution region.

Calculation:

We need to sketch the graph and label the vertices.

  

Hence, the result.

c.

To determine

Are the cholesterol levels within recommendations?

Answer to Problem 17PS

The given level is not well within the limit .

Explanation of Solution

Given information:

Cholesterol in human blood is necessary, but too much can lead to health problems. There are three main types of cholesterol: HDL (high-density lipoproteins), LDL (low-density lipoproteins), and VLDL (very low-density lipoproteins). HDL is considered “good” cholesterol; LDL and VLDL are considered “bad” cholesterol. A standard fasting cholesterol blood test measures total cholesterol, HDL cholesterol, and triglycerides. These numbers are used to estimate LDL and VLDL, which are difficult to measure directly. It is recommended that your combined LDL/VLDL cholesterol level be less than $130$ milligrams per decilitre, your HDL cholesterol level be at least $40$ milligrams per decilitre, and your total cholesterol level be no more than 200 milligrams per decilitre.

Are the following cholesterol levels within recommendations? Explain your reasoning.

LDL/VLDL: $120$ milligrams per decilitre

HDL: $90$ milligrams per decilitre

Total: $210$ milligrams per decilitre.

Calculation:

We need to check the following levels come under the recommendation limit or not.

LDL/VLDL: $120$ milligrams per decilitre

HDL: $90$ milligrams per decilitre

Total: $210$ milligrams per decilitre.

Hence, the given level is not well within the limit since the total cholesterol is greater than $200$ milligrams per decilitre.

d.

To determine

Give an example of cholesterol levels.

Answer to Problem 17PS

HDL − $135$ milligram per decilitre and LDL − $65$ milligram per decilitre and total cholesterol is $200$ milligram per decilitre.

Explanation of Solution

Given information:

Cholesterol in human blood is necessary, but too much can lead to health problems. There are three main types of cholesterol: HDL (high-density lipoproteins), LDL (low-density lipoproteins), and VLDL (very low-density lipoproteins). HDL is considered “good” cholesterol; LDL and VLDL are considered “bad” cholesterol. A standard fasting cholesterol blood test measures total cholesterol, HDL cholesterol, and triglycerides. These numbers are used to estimate LDL and VLDL, which are difficult to measure directly. It is recommended that your combined LDL/VLDL cholesterol level be less than $130$ milligrams per decilitre, your HDL cholesterol level be at least $40$ milligrams per decilitre, and your total cholesterol level be no more than $200$ milligrams per decilitre.

Give an example of cholesterol levels in which the LDL VLDL cholesterol level is too high but the HDL cholesterol level is acceptable.

Calculation:

We need to find a cholesterol level where LDL level is too high and HDL level and total cholesterol are in acceptable limits.

In order to make LDL in high level, make HDL level with some low acceptable level.

Let us take HDL level as $60$ , the LDL level can be $140$ milligrams per decilitre, which is too high and total cholestrol is $200$ milligram per decilitre.

Hence, another one can be HDL − $135$ milligram per decilitre and LDL − $65$ milligram per decilitre and total cholesterol is $200$ milligram per decilitre.

e.

To determine

Find a point in the solution region.

Answer to Problem 17PS

  $\left(75,105\right)$

Explanation of Solution

Given information:

Cholesterol in human blood is necessary, but too much can lead to health problems. There are three main types of cholesterol: HDL (high-density lipoproteins), LDL (low-density lipoproteins), and VLDL (very low-density lipoproteins). HDL is considered “good” cholesterol; LDL and VLDL are considered “bad” cholesterol. A standard fasting cholesterol blood test measures total cholesterol, HDL cholesterol, and triglycerides. These numbers are used to estimate LDL and VLDL, which are difficult to measure directly. It is recommended that your combined LDL/VLDL cholesterol level be less than $130$ milligrams per decilitre, your HDL cholesterol level be at least $40$ milligrams per decilitre, and your total cholesterol level be no more than $200$ milligrams per decilitre.

Another recommendation is that the ratio of total cholesterol to HDL cholesterol be less than 4 (that is, less than $4$ to $1$ ). Find a point in the solution region from part $\left(b\right)$ that meets this recommendation, and explain why it meets the recommendation

Calculation:

We have to find LDL and HDL level in such a way that the ratio of total cholesterol to HDL is less than $5$ . we can find so many answers for this.

Let us take 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 and the feasible region starts from the $x$ −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.

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