Rectangles with Changing Dimensions

Why Changing Dimensions Matter

A rectangle can change shape as its length and width change. Even when the dimensions change at different rates, the rectangle’s area and perimeter each have an instantaneous rate of change.

The key is to treat the length and width as quantities that depend on time. Differentiating the area and perimeter formulas with respect to time then connects the rates of the dimensions to the rates of the overall measurements.

Variables, Rates, and Units

Let \(L(t)\) be a rectangle’s length and \(W(t)\) its width at time \(t\). Both dimensions are assumed to be differentiable functions of time, and both remain positive while they describe a rectangle. Its area and perimeter are

A rate such as \(\frac{dL}{dt}\) describes how the length changes per unit of time at a particular instant. A positive rate means the dimension is increasing at that instant; a negative rate means it is decreasing. If lengths are measured in meters and time in minutes, then \(\frac{dL}{dt}\) has units of meters per minute, while \(\frac{dA}{dt}\) has units of square meters per minute.

The notation \(L'\), \(W'\), \(A'\), and \(P'\) in this tutorial means differentiation with respect to time. In particular, the prime on \(L\) here represents \(\frac{dL}{dt}\), not a derivative with respect to the length.

Differentiating Area and Perimeter

The area is a product of two time-dependent quantities. By the product rule, its rate is

Each term accounts for one side’s contribution to the area’s change. For example, if the width is momentarily fixed, only the first term remains. If the length is momentarily fixed, only the second term remains. When both sides change, both contributions must be included.

Perimeter is a sum of the two side lengths, each counted twice. Differentiating gives

These formulas relate instantaneous rates at the same time. To find an unknown side rate, use the dimensions and known rates at the instant in question, substitute them into the appropriate equation, and solve.

Why the Area Formula Uses the Product Rule

The product rule can be justified directly from the definition of a derivative. Suppose \(L\) and \(W\) are differentiable functions of time. For a small time change \(h\), the change in their product can be written by adding and subtracting \(L(t+h)W(t)\):

Divide both sides by \(h\), then take the limit as \(h\) approaches zero. Differentiability ensures that \(L(t+h)\) approaches \(L(t)\), and the difference quotients for \(L\) and \(W\) approach their derivatives. Therefore,

Since the rectangle’s area is \(A=LW\), this establishes the area-rate formula. The perimeter formula follows by differentiating each term of \(P=2L+2W\): the derivative of a constant multiple is the same constant times the derivative.

Finding an Unknown Side Rate from Area

If the area rate and both side lengths are known at an instant, the area equation can be rearranged to solve for either side rate. For instance, provided \(L\ne 0\),

The denominator is the current length, not a rate. This rearrangement is valid for a rectangle because its length is positive.

Finding an Unknown Side Rate from Perimeter

The perimeter equation is linear in the two side lengths. If the perimeter rate and one side rate are known, the other side rate follows by isolating it in \(\frac{dP}{dt}=2\frac{dL}{dt}+2\frac{dW}{dt}\). The factor of two remains in the rate equation because each changing side contributes to two sides of the rectangle’s boundary.

When Both Side Rates Are Known

Sometimes a problem gives both side rates and asks how quickly area or perimeter is changing. Substitute both rates into the relevant formula. The signs matter: a decreasing dimension contributes a negative term, while an increasing one contributes a positive term. The total can be positive, negative, or zero.

A Common Mistake: Losing the Factor of Two

A frequent error is to differentiate \(P=2L+2W\) as though \(P=L+W\). For example, in the perimeter example above, that incorrect calculation would give

This is not the stated perimeter rate. The perimeter includes two lengths and two widths, so each side rate must be counted twice:

A separate area error is to add the side rates directly. They have units of length per time, while an area rate has units of area per time. The product-rule terms \(W\frac{dL}{dt}\) and \(L\frac{dW}{dt}\) correctly produce area per time.

Choosing and Checking the Equation

Before solving, identify what quantity is changing and use its formula. For an area rate, use the product-rule equation; for a perimeter rate, use the sum with both factors of two. Then keep track of signed rates and units through the algebra. A negative answer for an unknown rate is meaningful: it says that side is decreasing at the instant.

A final substitution is an effective check. Put the calculated rate back into the differentiated relationship and verify that it reproduces the given rate. This catches sign errors, missing factors, and arithmetic mistakes.