While drilling a hole in the earth, it was found that the temperature `(T^(@)C)` at x km below the surface of the earth was given by T = 30 + 25(x - 3), when `3le xle15`.
Between which depths will the temperature be between `200^(@)C` and `300^(@)C`?

Answer

To solve the problem step by step, we need to find the range of depths (x) at which the temperature (T) is between 200°C and 300°C, given the temperature formula:

\[ T = 30 + 25(x - 3) \]

where \( 3 \leq x \leq 15 \).

### Step 1: Set up the inequality for temperature
We need to set up the inequalities for the temperature being between 200°C and 300°C:

\[ 200 < T < 300 \]

Substituting the expression for T into the inequality:

\[ 200 < 30 + 25(x - 3) < 300 \]

### Step 2: Break down the compound inequality
We can break this compound inequality into two separate inequalities:

1. \( 200 < 30 + 25(x - 3) \)
2. \( 30 + 25(x - 3) < 300 \)

### Step 3: Solve the first inequality
Let's solve the first inequality:

\[ 200 < 30 + 25(x - 3) \]

Subtract 30 from both sides:

\[ 170 < 25(x - 3) \]

Now, divide both sides by 25:

\[ \frac{170}{25} < x - 3 \]

Calculating \( \frac{170}{25} \):

\[ 6.8 < x - 3 \]

Adding 3 to both sides:

\[ 9.8 < x \]

### Step 4: Solve the second inequality
Now, let's solve the second inequality:

\[ 30 + 25(x - 3) < 300 \]

Subtract 30 from both sides:

\[ 25(x - 3) < 270 \]

Now, divide both sides by 25:

\[ x - 3 < \frac{270}{25} \]

Calculating \( \frac{270}{25} \):

\[ x - 3 < 10.8 \]

Adding 3 to both sides:

\[ x < 13.8 \]

### Step 5: Combine the results
Now we have two inequalities:

1. \( 9.8 < x \)
2. \( x < 13.8 \)

Combining these gives us:

\[ 9.8 < x < 13.8 \]

### Step 6: Consider the primary range
The primary range given in the problem is \( 3 \leq x \leq 15 \). The intersection of this primary range with our derived range \( 9.8 < x < 13.8 \) is:

\[ 9.8 < x < 13.8 \]

### Conclusion
Thus, the final answer is that the temperature will be between 200°C and 300°C for depths \( x \) in the range:

\[ (9.8, 13.8) \]