t the length of the side of square sheet of metal is increasing at the rate of  4cm/sec. the rate at which the area of the sheet is increasing whe length of the sides of a square sheet of metal is increasing athen the length of its side is 2cm

We are given that the length of the side of a square sheet of metal is increasing at the rate of 4 cm/sec. We need to find the rate at which the area of the sheet is increasing when the length of the sides is 2 cm.

### Step-by-step solution:

1. **Let the side of the square be** \( s \) (in cm).
   - The rate at which the side is increasing is: 
   \[
   \frac{ds}{dt} = 4 \, \text{cm/sec}
   \]

2. **The area** \( A \) of the square is given by the formula:
   \[
   A = s^2
   \]

3. **Differentiate the area** with respect to time \( t \) to find the rate of change of area:
   \[
   \frac{dA}{dt} = 2s \cdot \frac{ds}{dt}
   \]
   This equation relates the rate of change of the area \( \frac{dA}{dt} \) to the rate of change of the side \( \frac{ds}{dt} \).

4. **Substitute the given values**:
   - \( s = 2 \, \text{cm} \)
   - \( \frac{ds}{dt} = 4 \, \text{cm/sec} \)

   Now, calculate the rate of change of the area:
   \[
   \frac{dA}{dt} = 2(2) \cdot 4 = 16 \, \text{cm}^2/\text{sec}
   \]

Thus, the rate at which the area of the sheet is increasing when the side length is 2 cm is **16 cm²/sec**.

Would you like a more detailed explanation of any part of this process?

### Related questions:
1. What is the rate of change of the perimeter of the square in this case?
2. How would the rate of change of the area differ if the rate of change of the side length were 6 cm/sec?
3. If the side length was increasing at a constant rate, how would you find the total increase in area over a specific time?
4. How does the rate of change of the area depend on the size of the side of the square?
5. Can you generalize this process for a cube instead of a square, finding the rate of change of the volume?

### Tip:
In problems involving rates of change, always differentiate the key relationships (like area or volume formulas) with respect to time and then substitute the known rates and values.

The length of the side of a square sheet of metal is increasing at the rate of 4 cm/sec. What is the rate at which the area of the sheet is increasing when the length of its side is 2 cm?

Learn how to calculate the rate at which the area of a square sheet of metal increases when the side length grows at a constant rate. This problem involves differentiation and related rates, using key formulas like the area of a square and its rate of change. Suitable for high school students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation