The perimeter of a parallelogram with one internal angle `150^@` in 64 cm. Find the length of its sides when its area is maximum.

Options

A. 16, 16
B. 15, 17
C. 14, 18
D. can't be determined

Answer

To find the lengths of the sides of a parallelogram with a perimeter of 64 cm and one internal angle of 150° when its area is maximized, we can follow these steps:

### Step 1: Understand the properties of the parallelogram
A parallelogram has opposite sides that are equal in length. Let's denote the lengths of the sides as \( a \) and \( b \).

### Step 2: Write the equation for the perimeter
The perimeter \( P \) of a parallelogram is given by the formula:
\[ P = 2(a + b) \]
Given that the perimeter is 64 cm, we can set up the equation:
\[ 2(a + b) = 64 \]
Dividing both sides by 2 gives:
\[ a + b = 32 \quad \text{(Equation 1)} \]

### Step 3: Write the formula for the area
The area \( A \) of a parallelogram can be calculated using the formula:
\[ A = a \cdot b \cdot \sin(\theta) \]
where \( \theta \) is the angle between sides \( a \) and \( b \). Here, \( \theta = 150^\circ \).

### Step 4: Calculate \( \sin(150^\circ) \)
We know that:
\[ \sin(150^\circ) = \sin(180^\circ - 30^\circ) = \sin(30^\circ) = \frac{1}{2} \]
Thus, the area becomes:
\[ A = a \cdot b \cdot \frac{1}{2} = \frac{1}{2} ab \quad \text{(Equation 2)} \]

### Step 5: Substitute \( b \) from Equation 1 into Equation 2
From Equation 1, we can express \( b \) in terms of \( a \):
\[ b = 32 - a \]
Substituting this into the area formula gives:
\[ A = \frac{1}{2} a (32 - a) = \frac{1}{2} (32a - a^2) \]

### Step 6: Maximize the area
To maximize the area, we can differentiate \( A \) with respect to \( a \) and set the derivative to zero:
\[ A = \frac{1}{2} (32a - a^2) \]
Taking the derivative:
\[ \frac{dA}{da} = \frac{1}{2} (32 - 2a) \]
Setting the derivative equal to zero:
\[ 32 - 2a = 0 \implies 2a = 32 \implies a = 16 \]

### Step 7: Find \( b \)
Using \( a = 16 \) in Equation 1:
\[ b = 32 - a = 32 - 16 = 16 \]

### Conclusion
Thus, the lengths of the sides of the parallelogram when the area is maximized are:
\[ a = 16 \, \text{cm}, \, b = 16 \, \text{cm} \]