The problem presents four separate questions. We will solve question 3, related to a dam losing water. a) Find a formula for the function that describes the amount of water (in millions of liters) remaining in the dam each day. The dam has a capacity of 1150 million liters, and it loses 12 million liters daily. b) How long will it take for the dam to empty? c) When will the dam be half full?

1. Problem Description

The problem presents four separate questions. We will solve question 3, related to a dam losing water.

a) Find a formula for the function that describes the amount of water (in millions of liters) remaining in the dam each day. The dam has a capacity of 1150 million liters, and it loses 12 million liters daily.

b) How long will it take for the dam to empty?

c) When will the dam be half full?

2. Solution Steps

a) Let $f(t)$ be the amount of water in the dam (in millions of liters) after $t$ days.

The initial amount of water is 1150 million liters.

The dam loses 12 million liters per day, so the amount of water decreases by $12t$ after $t$ days.

Therefore, the function is $f(t) = 1150 - 12t$.

b) The dam is empty when $f(t) = 0$. So, we need to solve the equation $1150 - 12t = 0$ for $t$.

$12t = 1150$

$t = \frac{1150}{12} = \frac{575}{6} \approx 95.83$ days.

c) The dam is half full when it contains half of its capacity, which is $\frac{1150}{2} = 575$ million liters.

We need to find the time $t$ when $f(t) = 575$.

So, we solve the equation $1150 - 12t = 575$ for $t$.

$12t = 1150 - 575 = 575$

$t = \frac{575}{12} \approx 47.92$ days.

3. Final Answer

a) $f(t) = 1150 - 12t$

b) $\frac{575}{6} \approx 95.83$ days

c) $\frac{575}{12} \approx 47.92$ days