Textbook Question
Work done by a spring A spring on a horizontal surface can be stretched and held 0.5 m from its equilibrium position with a force of 50 N.
b. How much work is done in compressing the spring 0.5 m from its equilibrium position?
Ch. 6 - Applications of Integration
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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Table of contents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
Chapter 6, Problem 6.6.39b
In the design of solid objects (both artificial and natural), the ratio of the surface area to the volume of the object is important. Animals typically generate heat at a rate proportional to their volume and lose heat at a rate proportional to their surface area. Therefore, animals with a low SAV ratio tend to retain heat, whereas animals with a high SAV ratio (such as children and hummingbirds) lose heat relatively quickly.
b. What is the SAV ratio of a ball with radius a?
Verified step by step guidance1
Recall the formulas for the surface area and volume of a sphere (ball) with radius \(a\). The surface area \(S\) is given by \(S = 4 \pi a^2\).
The volume \(V\) of the sphere is given by \(V = \frac{4}{3} \pi a^3\).
The surface area to volume (SAV) ratio is defined as \(\frac{S}{V}\), so substitute the formulas for \(S\) and \(V\) into this ratio.
Write the SAV ratio as \(\frac{4 \pi a^2}{\frac{4}{3} \pi a^3}\) and simplify by canceling common factors such as \(4\) and \(\pi\).
After simplification, express the SAV ratio in terms of \(a\) only, which will show how the ratio depends on the radius of the ball.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Surface Area of a Sphere
The surface area of a sphere is calculated using the formula 4πa², where a is the radius. This represents the total area covering the outer layer of the ball, which is crucial for understanding how much heat can be lost through the surface.
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Example 1: Minimizing Surface Area
Volume of a Sphere
The volume of a sphere is given by the formula (4/3)πa³, where a is the radius. This measures the total space inside the ball, which relates to the amount of heat generated within the object.
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Surface Area to Volume (SAV) Ratio
The SAV ratio is the surface area divided by the volume, indicating how much surface is available per unit volume. For a sphere, this ratio helps explain heat retention or loss, as it compares the heat loss area to the heat-generating volume.
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09:07Example 1: Minimizing Surface Area
Related Practice
Textbook Question
Use the region R that is bounded by the graphs of y=1+√x,x=4, and y=1 complete the exercises.
Region R is revolved about the x-axis to form a solid of revolution whose cross sections are washers.
b. What is the inner radius of a cross section of the solid at a point x in [0, 4]?
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Textbook Question
Equal integrals Without evaluating integrals, explain the following equalities. (Hint: Draw pictures.)
b. ∫²₀(25−(x²+1)²) dx = 2∫₁⁵ y√y−1 dy
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Textbook Question
Use the region R that is bounded by the graphs of y=1+√x,x=4, and y=1 complete the exercises.
Region R is revolved about the y-axis to form a solid of revolution whose cross sections are washers.
b. What is the inner radius of a cross section of the solid at a point y in [1, 3]?
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Textbook Question
Probe speed A data collection probe is dropped from a stationary balloon, and it falls with a velocity (in m/s) given by v(t) = 9.8t, neglecting air resistance. After 10 s, a chute deploys and the probe immediately slows to a constant speed of 10 m/s, which it maintains until it enters the ocean.
b. How far does the probe fall in the first 30 s after it is released?
Textbook Question
Let R be the region bounded by the curve y=cos^−1x and the x-axis on [0, 1]. A solid of revolution is obtained by revolving R about the y-axis (see figures).
b. Find an expression for the area A(y) of a cross section of the solid at a point y in [0,π/2].