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Algebra and Trigonometry (MindTap ...

4th Edition
James Stewart + 2 others
ISBN: 9781305071742

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BuyFindarrow_forward

Algebra and Trigonometry (MindTap ...

4th Edition
James Stewart + 2 others
ISBN: 9781305071742
Textbook Problem

APPLICATIONS

Mixtures A chemist has three acid solutions at various concentrations. The first is 10% acid, the second is 20%, and the third is 40%. How many milliliters of each should she use to make 100 mL of 18% solution, if she has to use four times as much of the 10% solution as the 40% solution?

To determine

To find:

The number of milliliters required of three acid solutions.

Explanation

Given:

A chemist has three acid solutions at various concentrations. The first is 10% acid, the second is 20%, and the third is 40. If the chemist has to use four times as much of the 10% solution as the 40% solution

Approach:

Let x, y, and z represents the amount of 10%, 20%, and 40% solution.

Write the augmented matrix of the system.

Use elementary row operations to change the augmented matrix to row-echelon form.

Write the new system of equations that corresponds to the row-echelon form of the augmented matrix and solve by back-substitution.

Calculation:

The chemist used 18% of each solution to make 100 ml.

10%x+20%y+40%z=100ml of 18%10x+20y+40z=1800

Since the total amount of the solution is 100 ml.

Thus, x+y+z=100

The chemist uses four times as much of the 10% solution as the 40% solution.

Mathematically, x4z=0

The system of equations is given by,

10x+20y+40z=1800x+y+z=100x4z=0

Consider the system of equations {10x+20y+40z=1800x+y+z=100x4z=0.

The augmented matrix is given by,

[10204018001111001040]

Transform the augmented matrix into row echelon form.

Perform the transformation R1R110.

[1010201040101800101111001040][1241801111001040]

Perform the transformation R2R2R1.

[1241801112141001801040][124180013801040]

Perform the transformation R3R3R1.

[124180013801102440180][12418001380008180]

Perform the transformation R2R2

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