The sides of a square increase in length at a rate of $$ 6 \mathrm{~m} / \mathrm{sec} $$. a. At what rate is the area of the square changing when the sides are 11 m long? b. At what rate is the area of the square changing when the sides are 26 m long?

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To find the rate at which the area of the square is changing, we can use the relationship between the side length of the square and its area.

Let $$ s $$ be the length of a side of the square. The area $$ A $$ of the square is given by:

$$
A = s^2
$$

To find the rate of change of the area with respect to time, we can use the chain rule:

$$
\frac{dA}{dt} = \frac{dA}{ds} \cdot \frac{ds}{dt}
$$

First, we calculate $$ \frac{dA}{ds} $$:

$$
\frac{dA}{ds} = 2s
$$

Given that the sides of the square are increasing at a rate of $$ \frac{ds}{dt} = 6 \, \text{m/s} $$, we can substitute this into our equation:

$$
\frac{dA}{dt} = 2s \cdot \frac{ds}{dt} = 2s \cdot 6 = 12s
$$

Now we can find the rate of change of the area at the specified side lengths.

### a. When the sides are 11 m long:

Substituting $$ s = 11 \, \text{m} $$:

$$
\frac{dA}{dt} = 12 \cdot 11 = 132 \, \text{m}^2/\text{s}
$$

### b. When the sides are 26 m long:

Substituting $$ s = 26 \, \text{m} $$:

$$
\frac{dA}{dt} = 12 \cdot 26 = 312 \, \text{m}^2/\text{s}
$$

### Summary of Results:
- a. The area of the square is changing at a rate of $$ 132 \, \text{m}^2/\text{s} $$ when the sides are 11 m long.
- b. The area of the square is changing at a rate of $$ 312 \, \text{m}^2/\text{s} $$ when the sides are 26 m long.
a. The area of the square is changing at a rate of 132 $$ \text{m}^2/\text{s} $$ when the sides are 11 m long. b. The area of the square is changing at a rate of 312 $$ \text{m}^2/\text{s} $$ when the sides are 26 m long.

The sides of a square increase in length at a rate of \( 6 \mathrm{~m} / \mathrm{sec} \). a. At what rate is the area of the square changing when the sides are 11 m long? b. At what rate is the area of the square changing when the sides are 26 m long?

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