Ch. 3 - Derivatives

Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook

Hass 15th Edition

Ch. 3 - Derivatives

Problem 112a

Chapter 3, Problem 112a

How accurately should you measure the edge of a cube to be reasonably sure of calculating the cube’s surface area with an error of no more than 2%?

Verified step by step guidance

First, understand the formula for the surface area of a cube. The surface area \( S \) of a cube with edge length \( x \) is given by \( S = 6x^2 \).

Next, consider the concept of error propagation. We want the error in the surface area \( \Delta S \) to be no more than 2% of the actual surface area \( S \). This means \( \Delta S \leq 0.02S \).

To find \( \Delta S \), use the derivative of the surface area with respect to \( x \). The derivative \( \frac{dS}{dx} = 12x \) gives the rate of change of the surface area with respect to changes in \( x \).

The error in \( x \), denoted as \( \Delta x \), affects \( S \) through \( \Delta S \approx \frac{dS}{dx} \cdot \Delta x = 12x \cdot \Delta x \). Set this expression less than or equal to 2% of \( S \), i.e., \( 12x \cdot \Delta x \leq 0.02 \cdot 6x^2 \).

Solve the inequality \( 12x \cdot \Delta x \leq 0.12x^2 \) to find \( \Delta x \leq 0.01x \). This means the edge length \( x \) should be measured with an accuracy of at least 1% to ensure the surface area error is within 2%.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.

Video duration:

Was this helpful?

Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Surface Area of a Cube

The surface area of a cube is calculated using the formula A = 6s², where 's' is the length of one edge of the cube. Understanding this formula is essential because it directly relates the edge measurement to the total surface area, allowing us to assess how changes in 's' affect the calculated area.

Percentage Error

Percentage error is a way to express the accuracy of a measurement relative to the true value. It is calculated as (|measured value - true value| / true value) × 100%. In this context, we need to ensure that the error in measuring the edge of the cube leads to a surface area calculation error of no more than 2%, which requires careful consideration of how measurement inaccuracies propagate.

Differentiation and Error Propagation

Differentiation is a fundamental concept in calculus that helps us understand how a small change in one variable affects another variable. In this case, we can use differentiation to analyze how a small error in measuring the edge length 's' influences the surface area. This relationship is crucial for determining the maximum allowable error in the edge measurement to keep the surface area error within the specified 2% limit.

Recommended video:

04:57

Determining Error and Relative Error

Related Practice

Textbook Question

Find the linearizations of

a. tan x at x = -π/4

Graph the curves and linearizations together.

Textbook Question

The circumference of the equator of a sphere is measured as 10 cm with a possible error of 0.4 cm. This measurement is used to calculate the radius. The radius is then used to calculate the surface area and volume of the sphere. Estimate the percentage errors in the calculated values of

a. the radius.

b. the surface area.

c. the volume.

Textbook Question

To find the height of a lamppost (see accompanying figure), you stand a 6-ft pole 20 ft from the lamp and measure the length a of its shadow, finding it to be 15 ft, give or take an inch. Calculate the height of the lamppost using the value a = 15 and estimate the possible error in the result.

<IMAGE>

Textbook Question

Find the linearization of ƒ(x) = √(1 + x) + sin x - 0.5 at x = 0.

Textbook Question

Moving searchlight beam The figure shows a boat 1 km offshore, sweeping the shore with a searchlight. The light turns at a constant rate, dθ/dt = -0.6 rad/sec.

b. How many revolutions per minute is 0.6 rad/sec?

<IMAGE>