The length x of a rectangle is increasing at the rate of 3 cm/sec. and the width y is increasing at the rate of 2 cm/sec. If x=10 cm and y=6 cm, then the rate of change of its area is

Options

A. `2 cm^(2)//sec`
B. `-2 cm^(2)//sec`
C. `38 cm^(2)//sec`
D. `-38 cm^(2)//sec`

Answer

To find the rate of change of the area of a rectangle whose length and width are both increasing, we can follow these steps:

### Step 1: Define the variables
Let:
- \( x \) = length of the rectangle (in cm)
- \( y \) = width of the rectangle (in cm)
- \( A \) = area of the rectangle (in cm²)

### Step 2: Write the formula for the area
The area \( A \) of the rectangle is given by:
\[ A = x \cdot y \]

### Step 3: Differentiate the area with respect to time
To find the rate of change of the area with respect to time, we differentiate \( A \) using the product rule:
\[ \frac{dA}{dt} = y \cdot \frac{dx}{dt} + x \cdot \frac{dy}{dt} \]

### Step 4: Substitute the known values
From the problem, we know:
- \( \frac{dx}{dt} = 3 \) cm/sec (rate of change of length)
- \( \frac{dy}{dt} = 2 \) cm/sec (rate of change of width)
- At the given moment, \( x = 10 \) cm and \( y = 6 \) cm.

Now, substituting these values into the differentiated area equation:
\[ \frac{dA}{dt} = 6 \cdot 3 + 10 \cdot 2 \]

### Step 5: Calculate the values
Calculating each term:
- \( 6 \cdot 3 = 18 \)
- \( 10 \cdot 2 = 20 \)

Now, add these results together:
\[ \frac{dA}{dt} = 18 + 20 = 38 \text{ cm}^2/\text{sec} \]

### Final Answer
The rate of change of the area of the rectangle is:
\[ \frac{dA}{dt} = 38 \text{ cm}^2/\text{sec} \]

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