Problem 16 - Essential Calculus

Question

\mathrm{\\{} \text { Volume and Surface Area } The measurement of the edge of a cube is found to be 12 inches, with a possible error of 0.03 inch. Use differentials to approximate the maximum possible propagated error in computing (a) the volume of the cube and (b) the surface area of the cube.

Step-by-Step Solution

Answer

The maximum possible propagated error in (a) the volume is 12.96 cubic inches, and in (b) the surface area is 4.32 square inches.

Step 1: Formulate the volume and surface area of the cube

Let's denote the edge length of the cube by \(x\). So, \(x = 12\) inches. (a) The volume \(V\) of the cube is given by \(V = x^3\). (b) The surface area \(S\) of the cube is given by \(S = 6x^2\).

Step 2: Differentiate volume and surface area formulas

(a) Differentiate the volume formula: \(dV = 3x^2 dx\). (b) Differentiate the surface area formula: \(dS = 12x dx\).

Step 3: Compute the maximum possible propagated error

The possible error in edge measurement is given as 0.03 inch, which we denote as \(dx\). Substituting \(x = 12\) and \(dx = 0.03\) into the formulas: (a) \(dV = 3(12^2) (0.03)\) gives the propagated error in volume. (b) \(dS = 12(12) (0.03)\) gives the propagated error in surface area.

Key Concepts

Differentials

Differentials are a useful tool in calculus, especially when you want to approximate how a small change in one variable affects changes in other variables. They give us a way to estimate errors when measuring or calculating values.

When it comes to a cube, the edge length is a primary measurement. If we have a small change, denoted as \( dx \), this change will impact other properties, like volume and surface area. By using differentials, represented by \( dy \), we can predict how these properties are altered by \( dx \).

• For a function \( y(x) \), the differential \( dy \) is calculated as \( dy = f'(x)dx \), where \( f'(x) \) is the derivative of the function with respect to \( x \).
• This differential is a linear approximation of the change in \( y \) for a given change in \( x \).

Overall, differentials simplify the process of finding how small errors in primary measurements can lead to errors in computations involving those measurements.

Volume and Surface Area

In geometry, volume and surface area of a cube are key properties used extensively in applications:

• The volume \( V \) of a cube with edge length \( x \) is given by the formula \( V = x^3 \). This helps determine how much space is inside the cube.
• The surface area \( S \) is calculated by \( S = 6x^2 \), as there are six faces on a cube and each face has an area of \( x^2 \).

Understanding these formulas is crucial. They serve not only in theoretical mathematics but also in practical scenarios like packing and storage solutions.

Error Propagation

Error propagation is a method used to determine how errors affect derived measurements. When you know the amount of error in a measurement, you need to assess the impact this error can have on calculations derived from these measurements.

For the cube's volume and surface area:

• For volume, the propagated error is calculated by using \( dV = 3x^2 dx \). It estimates how inaccuracies in the edge length affect the volume.
• For surface area, use \( dS = 12x dx \). This formula assesses how small errors in the edge length measurement can influence the surface area.

Substituting known values:

• Plugging \( x = 12 \) inches and \( dx = 0.03 \) inch gives specific approximations of errors for the volume and surface area.

Using these approximations helps anticipate issues in structures or systems relying on exact measurements.