Question

1. a tank with 25 gallons drains at 10 gallons per hour. f(x) = -10x + 25 which is the correct domain for the situation?

Explanation:

Step1: Analyze the context

The function \( f(x)= - 10x + 25\) represents the amount of water (in gallons) in the tank after \(x\) hours. The amount of water can't be negative, and time \(x\) can't be negative.

Step2: Find the bounds for \(x\)

First, consider when the tank is empty: \(f(x)=0\). So, \(0=-10x + 25\). Solving for \(x\):

$$
LATEXBLOCK0
$$

Also, \(x\geq0\) (since time can't be negative) and \(f(x)\geq0\) (since the amount of water can't be negative), so \(x\leq2.5\).

Step3: Determine the domain

The domain of \(x\) (time in hours) should be from \(0\) to \(2.5\) (inclusive) because before \(x = 0\) (time hasn't started) and after \(x=2.5\) (tank is empty and can't have negative water), the function doesn't represent the real - world situation. So the domain is \(0\leq x\leq2.5\) (or in interval notation \([0, 2.5]\)).

Answer:

The domain of the function for this situation is all real numbers \(x\) such that \(0\leq x\leq2.5\) (or \([0, 2.5]\) in interval notation), where \(x\) represents the number of hours.