The sides of a square increase in length at a rate of
.
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?

Ask by Allan Byrd. in the United States

Oct 26,2024

Question

The sides of a square increase in length at a rate of
.
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?

Ask by Allan Byrd. in the United States

Oct 26,2024

UpStudy AI · Accepted Answer

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
.
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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The sides of a square increase in length at a rate of . 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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Latest Calculus Questions

The sides of a square increase in length at a rate of
.
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?