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 given, \( m = \frac{25}{x + 50} \), we need to determine the constraints on the variable \( m \).

1. Finding restrictions on \( m \):

   • As \( x \) (the amount of water in milliliters) increases, the denominator \( x + 50 \) also increases, which causes \( m \) to decrease.
   • When \( x \) is very small (approaching 0), the saline concentrate \( m \) approaches a maximum value, which occurs when \( x = 0\): \[ m = \frac{25}{0 + 50} = \frac{25}{50} = 0.5 \text{ (which corresponds to 50% concentration)} \]
   • As \( x \) increases towards infinity, \( m \) approaches 0: \[ \lim_{x \to \infty} m = \lim_{x \to \infty} \frac{25}{x + 50} = 0 \]
   • Therefore, \( m \) can take any value in the interval \( 0 < m \leq 0.5 \).

2. Filling in the blanks:

   • The value of \( m \) is restricted to positive numbers between 0 and 0.5.
   • The constraint on the variable \( m \) is written as 0 < m ≤ 0.5.

So the final answers are:

• The value of \( m \) is restricted to positive numbers between 0 and 0.5.
• The constraint on the variable \( m \) is written as 0 < m ≤ 0.5.