Write the maximum value of `12 sintheta-9sin^2\ \ thetadot`

Options

A. 3
B. 4
C. 5
D. 2

Answer

To find the maximum value of the expression \(12 \sin \theta - 9 \sin^2 \theta\), we can follow these steps:

### Step 1: Rewrite the expression
Let's denote \(y = 12 \sin \theta - 9 \sin^2 \theta\). We can rewrite this expression in terms of \(x = \sin \theta\):
\[ y = 12x - 9x^2 \]

### Step 2: Identify the quadratic form
The expression \(y = -9x^2 + 12x\) is a quadratic equation in the standard form \(y = ax^2 + bx + c\), where \(a = -9\), \(b = 12\), and \(c = 0\).

### Step 3: Find the vertex of the parabola
Since the coefficient of \(x^2\) is negative, the parabola opens downwards, and the maximum value occurs at the vertex. The x-coordinate of the vertex can be found using the formula:
\[ x = -\frac{b}{2a} \]
Substituting the values of \(a\) and \(b\):
\[ x = -\frac{12}{2 \times -9} = \frac{12}{18} = \frac{2}{3} \]

### Step 4: Substitute back to find the maximum value
Now, we substitute \(x = \frac{2}{3}\) back into the expression for \(y\):
\[ y = 12 \left(\frac{2}{3}\right) - 9 \left(\frac{2}{3}\right)^2 \]
Calculating each term:
\[ y = 12 \times \frac{2}{3} - 9 \times \frac{4}{9} \]
\[ y = 8 - 4 = 4 \]

### Conclusion
Thus, the maximum value of \(12 \sin \theta - 9 \sin^2 \theta\) is \(4\).