Q. 87 - Precalculus Enhanced with Graphing Utilities

Question

A doctor’s prescription calls for a daily intake of a supplement containing 40 milligrams (mg) of vitamin C and 30 mg of vitamin D. Your pharmacy stocks three supplements that can be used: one contains 20% vitamin C and 30% vitamin D; a second, 40% vitamin C and 20% vitamin D; and a third, 30% vitamin C and 50% vitamin D. Create a table showing the possible combinations that could be used to fill the prescription.

Step-by-Step Solution

Answer

The table of the combination of supplements is

Step 1. Given information

According to the doctor’s prescription, daily intake of supplements should contain $40$ milligrams (mg) of vitamin C and $30$ mg of vitamin D.

The first supplement contains $20$% vitamin  C and $30$% vitamin D

a second supplement contains $40$% vitamin C and $20$% vitamin D

a third supplement contains $30$% vitamin C and $50$% vitamin D

Step 2. Part (a)

Consider the quantity of the first compound  is x, the second compound is y, and the third compound is z

According to the doctor’s prescription, daily intake of supplements should contain  $40$milligrams (mg) of vitamin C 

so $0.2x+0.4y+0.3z=40 \cdots \left(i\right)$

According to the doctor’s prescription, daily intake of supplements should contain  $30$milligrams (mg) of vitamin D 

$0.3x+0.2y+0.5z=30 \cdots \left(ii\right)$

Step 3. Expression for y and x

Multiply equation i by $\frac{0.3}{0.2}$and subtract from equation ii

$$\begin{gathered} 0.3x+0.2y+0.5z=30 \\ \underline{-\left(0.3x+0.6y+0.45z=60\right)} \\ -0.4y+0.05z=-3 \end{gathered}$$

so $y=75+0.125z$

Multiply equation i by $0.5$and subtract from equation ii

$$\begin{gathered} 0.3x+0.2y+0.5z=30 \\ \underline{-\left(0.1x+0.2y+0.15z=20\right)} \\ 0.2x+0.35z=10 \end{gathered}$$

so$x=50-1.75z$

Step 4. Table of values

Substitute different values of z and determine values of x and y