Q. 88 - Precalculus Enhanced with Graphing Utilities

Question

A doctor’s prescription calls for the creation of pills that contain 12 units of vitamin $B_{12}$ and 12 units of vitamin E. Your pharmacy stocks three powders that can be used to make these pills: one contains 20% vitamin B12 and 30% vitamin E; a second, 40% vitamin $B_{12}$ and 20% vitamin E; and a third, 30% vitamin $B_{12}$ and 40% vitamin E. Create a table showing the possible combinations of each powder that could be mixed in each pill.

Step-by-Step Solution

Answer

The table showing the possible combinations of each powder

that could be mixed in each pill is

Step 1. Given information

According to the doctor’s prescription, the creation of pills should contain $12$units of vitamin $B_{12}$and $12$units of vitamin E.

The first powder contains $20$% vitamin $B_{12}$ and $30$% vitamin E

a second powder contains $40$% vitamin $B_{12}$ and $20$% vitamin E

a third powder contains $30$% vitamin $B_{12}$ and $40$% vitamin E

Step 2. Formation of equations

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

According to the doctor’s prescription, the creation of pills should contain $12$units of vitamin $B_{12}$

so  $$\begin{gathered} 0.2x+0.4y +0.3z=12 \\ 2x+4y+3z=120 \cdots \left(i\right) \end{gathered}$$

According to the doctor’s prescription, the creation of pills should contain $12$units of vitamin E.

So $$\begin{gathered} 0.3x+0.2y +0.4z=12 \\ 3x+2y+4z=120 \cdots \left(ii\right) \end{gathered}$$

Step 3. Expression for y and x

Multiply equation i by$\frac{3}{2}$ and subtract from equation ii

$$\begin{gathered} 3x+2y+4z =120 \\ \underline{-(3x+6y+4.5z=180)} \\ -4y-0.5z=-60 \end{gathered}$$

so $y=15-0.125z$

Divide equation i by $2$and subtract from equation ii

$$\begin{gathered} 3x+2y+4z =120 \\ \underline{-(1x+2y+1.5z=60)} \\ 2x+2.5z=60 \end{gathered}$$

so$x=30-1.25z$

Step 4. Table of values

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