the function m = 25

Question

x+50
gives the saline concentrate of a solution in milliliters when x milliliters of water is mixed with 50 milliliters of a 50% saline solution.
what are the constraints on the variable m in the function

the value of m is restricted to ''blank'' numbers
the constraint on the variable m is written as ''blank''

answer the blank spots with the correct answer.

Answer 1

To analyze the function \( m = \frac{25x + 50}{x + 50} \), we first consider the variables involved:

1. \( x \) represents the volume of water mixed in milliliters and must be greater than or equal to 0 (i.e., \( x \geq 0 \)).
2. The function measures the saline concentrate in milliliters, which must be positive and logically limited to maximum values based on the mixture.

To determine the constraints for \( m \):

• The minimum value occurs when \( x = 0 \): \[ m = \frac{25(0) + 50}{0 + 50} = \frac{50}{50} = 1 \]

• The maximum value occurs as \( x \) approaches infinity. As \( x \) becomes very large, the term \( \frac{25x}{x} \) approaches 25. Thus: \[ \text{As } x \to \infty, , m \to 25 \]

Therefore, the saline concentrate \( m \) will range from 1 to 25 (not inclusive), as we cannot physically have an infinite concentration of saline. The value of \( m \) is restricted to "positive numbers between 1 and 25".

The complete constraint on the variable \( m \) can be written as: "1 < m < 25".