We want to investigate the question: How much does the volume of a typical house increase over the course of a normal day's warming? This problem illustrates a number of basic ideas which are generalized in symbolic differentiation formulas, and the numerical answer may surprise you, besides. Do you think it increases by a thimble, a bucket, or a bathtub?

Most solids expand when they are heated in familiar temperature ranges. (Ice near freezing is a notable exception.) Scientific tables list "coefficients of expansion" in units of 1/(units of temperature). Here are a few in units of C:

The first question is: What does the "coefficient of expansion" mean? The units are actually (units of length expansion) per (units of length to start) per (units of temperature). The length units cancel. If the coefficient of expansion is c, the object is initially l units long, and the temperature increases C, then the change in length

Let's say the interior of the house is a rectangular solid 40 feet long, 30 feet wide, and 8 feet high. The changes in width and height are

How much does a gold palace the same size as your house expand its volume?

These numbers are interesting in that small linear expansions still produce a fairly big volume expansion, but all the arithmetic is not very revealing in terms of showing us the rates of expansion. A symbolic formulation can do this for us. We begin by specifying our variables:

where we begin with l₀=40, w₀=30, h₀=8, and V₀=9600 at the initial temperature, and c is a parameter for the coefficient of expansion. We know that V=lwh, but l, w, and h each depend on the temperature change . The coefficient of expansion formula gives us the formulas

so that the volume becomes

We can think of this expression as the composition of two simpler functions

the multiplier of a linear quantity needed to get its heated length and

The quantity x determines linear expansion, which in turn determines volume.

Give direct computations like those from Chapter 5 of the core text to show that a small change in volume as x changes to is

so that this change of produces the ratio or rate of change

Hint: Compute symbolically.

Now use the definition of the variable x.

Show that the change in x is as temperature changes from 0 to , so

Finally, consider the size of x² compared with the other terms.

Show that

and

Hint: Calculate x² numerically.

We wanted and approximated it in two steps, . The chain rule says in general that when V is "chained" together from two other functions.

Explain why the final expression

shows how the formula produces a large volume expansion from a small linear expansion, while at the same time the length of the house is only changed a tiny amount using the formula

Now give a "practical" application of your formula.

A worried homeowner hears of your computations and asks, "How cold would it have to get to contract my house's volume by 1%? I don't want to break the china." What is your answer?

7.1 Volume Expansion Explained by Calculus

Once you are comfortable with the symbolic rules of calculus, the large change in volume is easily understood. The instantaneous rate of thermal expansion of the length x of an object is the derivative . An object originally of unit length therefore has . Our house has dimensions l=40x, w=30x, and h=8x where the initial length x₀=1 foot. In terms of x, the volume of the house is

The chain rule says

and basic rules give

At the initial temperature x=1, thus we have or, in terms of differentials, the change in V is given by

Numerically, 3V₀c=.115, so that a change of temperature of dT=16 produces an approximate change in volume of

The insight into the large volume expansion is clearest in the initial symbolic formula for expansion

because the triple volume compensates for the small linear expansion coefficient, c.

Relative Expansion
While the linear expansion is small on our familiar scale, the volume seems larger - a bathtub full. However, it might be more reasonable to compare the change in volume to the amount we started with. Write an expression for the percentage change in volume and solve for that in terms of c and the change in temperature. Show that the percent change in volume is three times the percent change in length.

Problem 7.1

Do the scientific tables giving thermal expansion per unit length mean that

where x₀ is the fixed length at the initial temperature, while x is the varying length as temperature changes?

The first differential equation describes a straight line of constant slope x₀c, so the solution is x=x₀(1+cT). We will see in Chapter 8 of the core text that the solution to the second equation is x=x₀e^cT. Use Mathematicato plot both functions over the range with x₀=1 and . What do you observe? In particular, what is the largest difference between (1+cT) and e^cT over this range?

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