RELATED RATES – Square Problem

Each side of a square is increasing at a rate of 6 $\frac{cm}{s}$. At what rate is the area of the square increasing when the area of the square is 16 $cm^2$?

If you haven’t already, you should check out my related rates lesson. I go through the steps that can be used to solve any related rates problem. I will use these steps here, but it might be useful for you to see a more detailed explanation of why we use these steps.

1. Draw a sketch

The first thing we will always want to do is draw a sketch of the situation described by the problem. This problem is relatively simple to draw out. Here we have a square whose sides are growing at a constant rate. We know the sides are growing at a rate of 6 $\frac{cm}{s}$ and we want to consider the moment when the area of the square is 16 $cm^2$.

2. Come up with your equation

Now that we have our drawing laid out, we need to create an equation whose derivative we will need to find.

What are we looking for?

This question is asking us to find the rate at which the area of the square is increasing. So it tells us that we need to have some variable that represents the rate of change of the area at some point.

Of course, we will need to take the derivative of our equation soon. Doing this will introduce the ‘rate of change’ part. Therefore, we need to make sure that our equation we make includes the area of the square. As long as we do this, we will end up with the rate of change of the area once we take its derivative.

There was only two pieces of information that the question directly told us.

• Each side of a square is increasing at a rate of 6 $\frac{cm}{s}$.
• The area of the square in this instant is 16 $cm^2$.

Putting it into an equation

So at this point, we have figured out that we need our equation to include the square’s area, and we know something about the square’s area and the rate of change of its sides.

Although, we don’t know anything about the actual side lengths at the given instant, we can figure that out if we have to. Therefore, it’s fine if our equation includes the length of the square’s sides.

To summarize, the only measurements of this square we have discussed so far are its side lengths and its area. So our equation should relate its area and its side lengths. Keep in mind, we don’t want to put anything that represents a rate of change into our equation. These will come when we take the derivative of our equation.

The simplest equation that relates the area of a square with its side lengths would likely be the formula for the area of a square that you are already familiar with. $$A=l^2$$ Where A is the area of the square, and l is the length of its sides.

3. Implicit differentiation

Once we have created our equation like we did above, we need to go ahead and take its derivative with respect to time. This will be done using implicit differentiation.

Since we will be taking the derivative with respect to time, we will need to treat the A and the l in our equation as functions of time. This will require using the chain rule to find their derivatives.

$$\frac{d}{dt} \big[ A \big] = \frac{d}{dt} \Big[ l^2 \Big]$$ $$\frac{dA}{dt} = 2l \cdot \frac{dl}{dt}$$

4. Solve for the desired rate of change

Now all we have to do is solve for the rate of change the question is asking about. Just like I said earlier, we need to find the rate of change of the square’s area. Since this is exactly what $\frac{dA}{dt}$ represents and we have already isolated this, we don’t have much else to do. All we need to do is plug in the values we have for everything else in our equation.

The only other variables we need to plug in are l and $\frac{dl}{dt}$. The tricky thing here is that we don’t directly know what l is at the moment in this problem. Instead we know that the area of the square is
16 $cm^2$. Since we know that the relationship between the area of a square and its side lengths is $$A=l^2$$ we can find l in this instant. $$16=l^2$$ $$4=l$$ So we know that in this moment the length of the square’s sides is 4 cm.

Remember, the problem also told us that the side lengths are increasing at a rate of 6 $\frac{cm}{s}$. This directly tells us the rate of change of the sides lengths. So we also know that $$\frac{dl}{dt}=6$$

Putting all of this into our equation will give us: $$\frac{dA}{dt} = 2l \cdot \frac{dl}{dt}$$ $$\frac{dA}{dt} = 2(4)(6)$$ $$\frac{dA}{dt} = 48$$

So the area of this square is increasing at a rate of 48 $\mathbf{\frac{cm^2}{s}}$ when the area is 16 $\mathbf{cm^2}$.

If you’re still having some trouble with related rates problems or just want some more practice you should check out my related rates lesson. At the bottom of this lesson there is a list of related rates problems that I have posted along with their solutions. I also have several other lessons and problems on the derivatives page you can check out. If you can’t find the topic or question you’re looking for just let me know by emailing me at jakesmathlessons@gmail.com!

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