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CalculusFinite Mathematics and Applied Calculus (MindTap Course List)More Balloons A rather flimsy spherical balloon is designed to pop at the instant its radius has reached 10 centimeters. Assuming that the balloon is filled with helium at a rate of 10 cm 3 /sec , calculate how fast the radius is growing at the instant it pops. [ HINT: See Example 1. (The volume of a sphere of radius r is V = 4 3 π r 3 .)]BuyFind*arrow_forward*

7th Edition

Stefan Waner + 1 other

Publisher: Cengage Learning

ISBN: 9781337274203

Chapter 12.5, Problem 12E

Textbook Problem

**More Balloons** A rather flimsy spherical balloon is designed to pop at the instant its radius has reached 10 centimeters. Assuming that the balloon is filled with helium at a rate of
**HINT:** See Example 1. (The volume of a sphere of radius *r* is

Expert Solution

To determine

**To calculate:** The rate at which the radius is growing at instant it pops where a rather filmy spherical balloon is designed to pop at the instant its radius has reached

**Given information:**

It is provided that a rather filmy spherical balloon is designed to pop at the instant its radius has reached

**Formula used:**

The steps for solving the related rate problem are:

A. The Question

1. List the quantities that are related and changed with time.

2. Restate the question in the form of rate of change and write again the question using changing quantities for their derivatives.

B. The Relationship

1. Make a diagram that shows the changing quantities.

2. Determine an equation or equations that relates the changing quantities.

3. Determine the derivative with respect to the equation(s) that relates the quantity to find derived equation(s), which relate the rate of change of the quantities.

C. The answer

1. Make a substitution to the derived equation(s), the given values of the quantities and their derivatives.

2. Calculate the required derivative.

Volume of the sphere:

The Volume of the sphere whose radius is

**Calculation:**

As it is provided that a rather filmy spherical balloon is designed to pop at the instant its radius has reached

Consider the steps of the formula:

Step 1: The problem

The changing quantities are volume

Since, the balloon is filled with helium at a rate of

Thus,

The rate of change of volume is

Thus,

Since, a rather filmy spherical balloon is designed to pop at the instant its radius has reached

Finite Mathematics and Applied Calculus (MindTap Course List)

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If you know how fast one quantity is changing and...Ch. 12.5 - In a related rates problem there is no limit to...Ch. 12.5 - If three quantities are related by a single...Ch. 12.5 - The demand and unit price for your stores...Ch. 12.5 - The number N of employees and the total floor...Ch. 12.5 - In solving a related rates problem a key step is...Ch. 12.5 - On a recent exam you were given a related rates...Ch. 12.5 - Transform the following into a mathematical...Ch. 12.5 - If two quantities x and y are related by a linear...Ch. 12.6 - Demand for Oranges The weekly sales of Honolulu...Ch. 12.6 - Demand for Oranges Repeat Exercise 1 for weekly...Ch. 12.6 - Demand for Smartphones Worldwide annual sales of...Ch. 12.6 - Projected Revenue: Smartphones Worldwide annual...Ch. 12.6 - College Tuition An old study of about 1,800 U.S....Ch. 12.6 - Monorail Services The demand for monorail service...Ch. 12.6 - Tissues The consumer demand equation for tissues...Ch. 12.6 - Bodybuilding The consumer demand curve for...Ch. 12.6 - T-Shirts The Physics Club sells E=mc2 T-shirts at...Ch. 12.6 - Comics The demand curve for original Iguanawoman...Ch. 12.6 - E-Readers The demand for Amazons Kindle e-reader...Ch. 12.6 - Monorail Service on Mars The demand for monorail...Ch. 12.6 - Corn In the 1930s the economist Henry Schultz...Ch. 12.6 - Demand for Fried Chicken A fried chicken franchise...Ch. 12.6 - Paint-By-Number The estimated monthly sales of...Ch. 12.6 - Paint-By-Number Repeat Exercise 15 using the...Ch. 12.6 - Linear Demand Functions A general linear demand...Ch. 12.6 - Exponential Demand Functions A general exponential...Ch. 12.6 - Hyperbolic Demand Functions A general hyperbolic...Ch. 12.6 - Quadratic Demand Functions A general quadratic...Ch. 12.6 - Modeling Linear Demand You have been hired as a...Ch. 12.6 - Modeling Linear Demand You have been hired as a...Ch. 12.6 - Modeling Exponential Demand As the new owner of a...Ch. 12.6 - Modeling Exponential Demand Repeat Exercise 23,...Ch. 12.6 - Income Elasticity of Demand: Live Drama The...Ch. 12.6 - Income Elasticity of Demand: Live Concerts The...Ch. 12.6 - Income Elasticity of Demand: Broadband in 2010 The...Ch. 12.6 - Income Elasticity of Demand: Broadband in 2007 The...Ch. 12.6 - Income Elasticity of Demand: Computer Usage in the...Ch. 12.6 - Income Elasticity of Demand: Internet Usage in the...Ch. 12.6 - Price Elasticity of Supply Given a supply equation...Ch. 12.6 - Price Elasticity of Supply Given a supply equation...Ch. 12.6 - Income Elasticity of Demand (based on a question...Ch. 12.6 - College Tuition (from the GRE Economics Test) A...Ch. 12.6 - Complete the following: When demand is inelastic,...Ch. 12.6 - Complete the following: When demand has unit...Ch. 12.6 - Given that the demand q is a differentiable...Ch. 12.6 - Given that the demand q is a differentiable...Ch. 12.6 - Your calculus study group is discussing price...Ch. 12.6 - Another member of your study group claims that...Ch. 12 - In Exercises 18, find all the relative and...Ch. 12 - In Exercises 18, find all the relative and...Ch. 12 - In Exercises 18, find all the relative and...Ch. 12 - In Exercises 18, find all the relative and...Ch. 12 - In Exercises 18, find all the relative and...Ch. 12 - In Exercises 18, find all the relative and...Ch. 12 - In Exercises 18, find all the relative and...Ch. 12 - In Exercises 18, find all the relative and...Ch. 12 - In Exercises 912, find the approximate...Ch. 12 - In Exercises 912, find the approximate...Ch. 12 - In Exercises 912, find the approximate...Ch. 12 - In Exercises 912, find the approximate...Ch. 12 - In Exercises 13 and 14, f is continuous on [3,3],...Ch. 12 - In Exercises 13 and 14, f is continuous on [3,3],...Ch. 12 - In Exercises 15 and 16 the position s of a point...Ch. 12 - In Exercises 15 and 16 the position s of a point...Ch. 12 - In Exercises 1722, sketch the graph of the given...Ch. 12 - In Exercises 1722, sketch the graph of the given...Ch. 12 - In Exercises 1722, sketch the graph of the given...Ch. 12 - In Exercises 1722, sketch the graph of the given...Ch. 12 - In Exercises 1722, sketch the graph of the given...Ch. 12 - In Exercises 1722, sketch the graph of the given...Ch. 12 - Revenue Demand for the latest best-seller at...Ch. 12 - Revenue Demand for The Secret Loves of John O, a...Ch. 12 - Profit Taking into account storage and shipping,...Ch. 12 - Profit Taking into account storage and shipping,...Ch. 12 - Office Space Although still a sophomore at...Ch. 12 - Recreation Space As a result of complaints by the...Ch. 12 - Box Design The sales department at...Ch. 12 - Box Redesign The sales department at...Ch. 12 - Sales OHaganBooks.com modeled its weekly sales...Ch. 12 - Sales The quarterly sales of OHagan oPods...Ch. 12 - Chance Encounter Marjory Duffin is walking north...Ch. 12 - Company Logos OHaganBooks.coms website has an ani-...Ch. 12 - Elasticity of Demand (Compare Exercise 23.) Demand...Ch. 12 - Elasticity of Demand (Compare Exercise 24.) Demand...Ch. 12 - Elasticity of Demand Last year OHaganBooks.com...Ch. 12 - Elasticity of Demand JungleBooks.com (one of...

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