Q.In drilling world's deepest hole it was found that the temperature $T$ in degree celcius, $x$ km below the earth's surface was given by $T = 30 + 25(x - 3)$, $3 \le x \le 15$. At what depth will the temperature be between $155^\circ\text{C}$ and $205^\circ\text{C}$?

The problem requires finding the range of depths for a given temperature range, which involves solving a linear inequality.

The temperature $T$ is given by $T = 30 + 25(x - 3)$, where $x$ is the depth in km and $3 \le x \le 15$.

We need to find $x$ when the temperature is between $155^\circ\text{C}$ and $205^\circ\text{C}$. This translates to the inequality:

Substitute the expression for $T$:

Subtract 30 from all parts of the inequality:

Divide all parts by 25:

$5 < x - 3 < 7$ …

When we are given a formula that relates two quantities, like temperature and depth, and we need to find the range of one quantity (depth) for a given range of the other (temperature), we use inequalities. The core idea is to set up an inequality that reflects the desired condition for temperature, then use algebraic manipulation to isolate the variable representing depth.

The key here is to remember that when you perform operations on an inequality, you must do it to all parts of the inequality. Also, be careful with multiplying or dividing by negative numbers, as this flips the inequality signs. In this problem, we will only be dealing with positive numbers for multiplication and division, so the signs will remain as they are. Finally, we must always check our derived range against any given domain constraints for the variable.

Let's break down the solution step-by-step.

1. Understand the given information: We are given the formula for temperature $T$ in degrees Celsius at a depth $x$ km below the Earth's surface:

This formula is valid for depths $x$ such that $3 \le x \le 15$.
We need to find the depth $x$ where the temperature $T$ is between $155^\circ\text{C}$ and $205^\circ\text{C}$. This means $T$ must be strictly greater than $155^\circ\text{C}$ and strictly less than $205^\circ\text{C}$.

2. Formulate the inequality for temperature:

The condition "temperature will be between $155^\circ\text{C}$ and $205^\circ\text{C}$" translates to the compound inequality:

3. Substitute the temperature formula into the inequality: Now, we replace $T$ with its given expression $30 + 25(x - 3)$:

This is a compound linear inequality that we need to solve for $x$.

4. Isolate the term containing $x$ (the $25(x-3)$ part):

To isolate $25(x - 3)$, we first need to subtract $30$ from all three parts of the inequality. This operation does not change the direction of the inequality signs.

Next, we divide all three parts by $25$. Since $25$ is a positive number, the inequality signs remain unchanged.

5. Isolate $x$: To get $x$ by itself, we add $3$ to all three parts of the inequality. This operation also does not change the direction of the inequality signs. …