Find the minimum and maximum value of `7cosalpha+24sinbeta`.

Options

A. `-7and7`
B. `-25and25`
C. `-24and24`
D. `-31and31`

Answer

To find the minimum and maximum values of the expression \(7 \cos \alpha + 24 \sin \beta\), we can use the following approach:

### Step-by-Step Solution:

1. **Identify the coefficients**:
In the expression \(7 \cos \alpha + 24 \sin \beta\), we can identify:
- \(a = 24\) (coefficient of \(\sin \beta\))
- \(b = 7\) (coefficient of \(\cos \alpha\))
- \(c = 0\) (since there is no constant term added)

2. **Use the formula for maximum and minimum values**:
The maximum and minimum values of the expression \(a \cos \alpha + b \sin \beta + c\) can be found using the formulas:
- Maximum value = \(c + \sqrt{a^2 + b^2}\)
- Minimum value = \(c - \sqrt{a^2 + b^2}\)

3. **Calculate \(a^2 + b^2\)**:
\[ a^2 = 24^2 = 576 \]
\[ b^2 = 7^2 = 49 \]
\[ a^2 + b^2 = 576 + 49 = 625 \]

4. **Find the square root**:
\[ \sqrt{a^2 + b^2} = \sqrt{625} = 25 \]

5. **Calculate the maximum value**:
\[ \text{Maximum value} = c + \sqrt{a^2 + b^2} = 0 + 25 = 25 \]

6. **Calculate the minimum value**:
\[ \text{Minimum value} = c - \sqrt{a^2 + b^2} = 0 - 25 = -25 \]

### Final Result:
- The maximum value of \(7 \cos \alpha + 24 \sin \beta\) is **25**.
- The minimum value of \(7 \cos \alpha + 24 \sin \beta\) is **-25**.