Find the maximum and minimum values of `7 cos theta+24 sin theta:`

Options

A. `25 and -25`
B. `16 and-9`
C. `25 and 0`
D. `36 and 25`

Answer

To find the maximum and minimum values of the expression \(7 \cos \theta + 24 \sin \theta\), we can use the formula for the maximum and minimum values of the expression of the form \(a \cos \theta + b \sin \theta\).

### Step-by-Step Solution:

1. **Identify coefficients**:
Here, we have \(a = 7\) and \(b = 24\).

2. **Calculate the maximum value**:
The maximum value of \(a \cos \theta + b \sin \theta\) is given by the formula:
\[ \text{Maximum value} = \sqrt{a^2 + b^2} \]
Plugging in the values:
\[ \text{Maximum value} = \sqrt{7^2 + 24^2} \]

3. **Calculate \(7^2\) and \(24^2\)**:
\[ 7^2 = 49 \]
\[ 24^2 = 576 \]

4. **Sum the squares**:
\[ 49 + 576 = 625 \]

5. **Take the square root**:
\[ \sqrt{625} = 25 \]
Therefore, the maximum value is \(25\).

6. **Calculate the minimum value**:
The minimum value of \(a \cos \theta + b \sin \theta\) is given by:
\[ \text{Minimum value} = -\sqrt{a^2 + b^2} \]
Thus, the minimum value is:
\[ \text{Minimum value} = -\sqrt{7^2 + 24^2} = -25 \]

### Final Results:
- Maximum value: \(25\)
- Minimum value: \(-25\)