# Question Video: Finding the Rate of Change in the Area of an Expanding Square-Shaped Lamina Using Related Rates Mathematics • Higher Education

A square lamina maintains its shape as it expands. Find the rate of change in its area with respect to side length when its side measures 4 cm.

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### Video Transcript

A square lamina maintains its shape as it expands. Find the rate of change in its area with respect to side length when its side measures four centimeters.

We have a square lamina, which is a two-dimensional flat object whose thickness is negligible. We’re asked to find the rate of change of its area with respect to side length. And that’s given a particular side length of four centimeters. Let’s begin by calling the side length 𝑥. Then we can find an equation for the area with respect to side length. Since our lamina is a square and we know that the area of a square is the square of the side length, as a function of side length 𝑥, the area 𝐴 of 𝑥 is equal to 𝑥 squared.

Now, normally, when we think of rates of change, we think of the rate of change with respect to time. But we’re not given any information about the rate of change with respect to time. All we have is the area as a function of side length.

To find the rate of change with respect to side length then, we differentiate the area function with respect to side length 𝑥. That is, we find d𝐴 by d𝑥. And to do this, we use the power rule for derivatives. This tells us that for any real power 𝑛 and constant coefficient 𝑎, d by d𝑥 of the function 𝑎 times 𝑥 to the power 𝑛 is 𝑛 multiplied by 𝑎 times 𝑥 raised to the power 𝑛 minus one. That is, we bring the power 𝑛 down to the front and multiply by it and subtract one from the power or exponent. In our case, this means bringing down the exponent two and multiplying by it. And then our new exponent is two minus one. This is one, and since anything to the power one is simply itself, we have d𝐴 by d𝑥 equals two 𝑥.

Now, to find the rate of change in the lamina’s area when its side length is four, we substitute four for 𝑥 in d𝐴 by d𝑥. So we have two times four, which is eight. Since our area is measured in square centimeters and our side length in centimeters, the units are square centimeters per centimeter. That is the number of square centimeters resulting from a one-centimeter change in side length. Hence, the rate of change of the lamina’s area with respect to side length when its side measures four centimeters is eight square centimeters per centimeter.