Maximum value of `24 sin theta+7 cos theta` is

Options

A. `7`
B. `17`
C. `24`
D. `25`

Answer

To find the maximum value of the expression \( 24 \sin \theta + 7 \cos \theta \), we can use the formula for the maximum value of the expression \( a \sin \theta + b \cos \theta \), which is given by:

\[ \text{Maximum Value} = \sqrt{a^2 + b^2} \]

### Step-by-Step Solution:

**Step 1: Identify the coefficients.**
- Here, \( a = 24 \) and \( b = 7 \).

**Step 2: Calculate \( a^2 \) and \( b^2 \).**
- Calculate \( a^2 = 24^2 = 576 \).
- Calculate \( b^2 = 7^2 = 49 \).

**Step 3: Add \( a^2 \) and \( b^2 \).**
- Now, add these two values:
\[ a^2 + b^2 = 576 + 49 = 625 \]

**Step 4: Take the square root.**
- Finally, take the square root of the sum:
\[ \sqrt{625} = 25 \]

Thus, the maximum value of \( 24 \sin \theta + 7 \cos \theta \) is \( 25 \).

### Final Answer:
The maximum value is \( 25 \).

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