What is the maximum value of `16 sin theta -12 sin^(2)theta` ?

Options

A. A)`3//4`
B. B)`4//3`
C. C)`16//3`
D. D)4

Answer

To find the maximum value of the expression \( y = 16 \sin \theta - 12 \sin^2 \theta \), we can follow these steps:

### Step 1: Rewrite the expression
Let \( x = \sin \theta \). Then, we can rewrite the expression as:
\[ y = 16x - 12x^2 \]

### Step 2: Differentiate the expression
To find the maximum value, we need to find the derivative of \( y \) with respect to \( x \):
\[ \frac{dy}{dx} = 16 - 24x \]

### Step 3: Set the derivative to zero
To find the critical points, set the derivative equal to zero:
\[ 16 - 24x = 0 \]

### Step 4: Solve for \( x \)
Now, solve for \( x \):
\[ 24x = 16 \\ x = \frac{16}{24} = \frac{2}{3} \]

### Step 5: Find the maximum value of \( y \)
Now, substitute \( x = \frac{2}{3} \) back into the expression for \( y \):
\[ y = 16\left(\frac{2}{3}\right) - 12\left(\frac{2}{3}\right)^2 \]
Calculating each term:
\[ y = \frac{32}{3} - 12 \cdot \frac{4}{9} \]
\[ y = \frac{32}{3} - \frac{48}{9} \]
Convert \(\frac{48}{9}\) to have a common denominator of 3:
\[ \frac{48}{9} = \frac{16}{3} \]
So we have:
\[ y = \frac{32}{3} - \frac{16}{3} = \frac{16}{3} \]

### Conclusion
The maximum value of \( 16 \sin \theta - 12 \sin^2 \theta \) is:
\[ \boxed{\frac{16}{3}} \]