Logout Open In App
Open In App
Warning: foreach() argument must be of type array|object, bool given in /var/www/html/web/app/themes/studypress-core-theme/template-parts/header/mobile-offcanvas.php on line 20

Chapter 4: Problem 51

Minting Coins A manufacturer contracts to mint coins for the federal government. The coins must weigh within 0.1\(\%\) of their ideal weight, so the volume must be within 0.1\(\%\) of the ideal volume. Assuming the thickness of the coins does not change, what is the percentage change in the volume of the coin that would result from a 0.1\(\%\) increase in the radius?

Short Answer

Expert verified
The percentage change in the volume of the coin resulting from a 0.1\(\%\) increase in the radius is approximately 0.2\(\%\)

Step by step solution

01

Define the Formula of Volume

Since the coins can be treated as cylinders, the formula to determine their volume can be given as \( V = \pi*r^{2}*h \) where \( r \) is the radius and \( h \) is the height (which is referred to as thickness in this problem). Note that the height does not change.
02

Calculate Percentage Change in Volume with Change in Radius

The volume of the coin increases with the square of the radius. Therefore, a 0.1\(\%\) increase in radius would result in approximately a 2*0.1\(\%\) = 0.2\(\%\) increase in the volume when other parameters remain constant (based on the approximation that the change in \( r^{2} = r^{2} * \Delta \%r \) ).
03

Conclusion

So, a 0.1\(\%\) increase in the radius of the coins causes an approximate 0.2\(\%\) increase in the volume of coins. This change in volume is ideally within the acceptable range of 0.1\(\%\) as mentioned in the problem

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Multiple Choice If the volume of a cube is increasing at 24 \(\mathrm{in}^{3} / \mathrm{min}\) and the surface area of the cube is increasing at 12 \(\mathrm{in}^{2} / \mathrm{min}\) , what is the length of each edge of the cube? \(\mathrm{}\) \(\begin{array}{lll}{\text { (A) } 2 \text { in. }} & {\text { (B) } 2 \sqrt{2} \text { in. (C) } \sqrt[3]{12} \text { in. (D) } 4 \text { in. }}\end{array}\)

Production Level Show that if \(r(x)=6 x\) and \(c(x)=\) \(x^{3}-6 x^{2}+15 x\) are your revenue and cost functions, then the best you can do is break even (have revenue equal cost).

Newton's Method Suppose your first guess in using Newton's method is lucky in the sense that \(x_{1}\) is a root of \(f(x)=0 .\) What happens to \(x_{2}\) and later approximations?

$$ \begin{array}{l}{\text { Multiple Choice If } f(x)=\cos x, \text { then the Mean Value }} \\ {\text { Theorem guarantees that somewhere between } 0 \text { and } \pi / 3, f^{\prime}(x)=} \\ {\text { (A) }-\frac{3}{2 \pi} \quad \text { (B) }-\frac{\sqrt{3}}{2} \quad(\mathbf{C})-\frac{1}{2} \quad \text { (D) } 0}\end{array} $$

Measuring Acceleration of Gravity When the length \(L\) of a clock pendulum is held constant by controlling its temperature, the pendulum's period \(T\) depends on the acceleration of gravity \(g\) . The period will therefore vary slightly as the clock is moved from place to place on the earth's surface, depending on the change in \(g\) . By keeping track of \(\Delta T\) , we can estimate the variation in \(g\) from the equation \(T=2 \pi(L / g)^{1 / 2}\) that relates \(T, g,\) and \(L .\) (a) With \(L\) held constant and \(g\) as the independent variable, calculate \(d T\) and use it to answer parts \((b)\) and \((c)\) . (b) Writing to Learn If \(g\) increases, will \(T\) increase or decrease? Will a pendulum clock speed up or slow down? Explain. (c) A clock with a 100 -cm pendulum is moved from a location where \(g=980 \mathrm{cm} / \mathrm{sec}^{2}\) to a new location. This increases the period by \(d T=0.001 \mathrm{sec} .\) Find \(d g\) and estimate the value of \(g\) at the new location.
See all solutions

Recommended explanations on Math Textbooks

Discrete Mathematics

Read Explanation

Mechanics Maths

Read Explanation

Pure Maths

Read Explanation

Statistics

Read Explanation

Calculus

Read Explanation

Geometry

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.

Sign-up for free