Textbook Question
Surface area of a cone The lateral surface area of a cone of radius r and height h (the surface area excluding the base) is A = πr√r²+h².
a. Find dr/dh for a cone with a lateral surface area of A=1500π.
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Ch. 3 - Derivatives
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 3, Problem 3.8.44a
Volume of a torus The volume of a torus (doughnut or bagel) with an inner radius of a and an outer radius of b is V=π²(b+a)(b−a)²/4.
a. Find db/da for a torus with a volume of 64π².
Verified step by step guidance1
Start by understanding the given formula for the volume of a torus: V = π²(b+a)(b−a)²/4. We need to find the derivative db/da when the volume V is 64π².
Set the volume equation equal to 64π²: π²(b+a)(b−a)²/4 = 64π². Simplify this equation to find a relationship between a and b.
Multiply both sides of the equation by 4/π² to isolate the terms involving a and b: (b+a)(b−a)² = 256.
To find db/da, implicitly differentiate both sides of the equation with respect to a. Use the product rule and chain rule where necessary.
Solve the resulting equation for db/da, ensuring to substitute any known values or relationships between a and b from the previous steps.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Volume of a Torus
The volume of a torus is calculated using the formula V = π²(b + a)(b - a)²/4, where 'a' is the inner radius and 'b' is the outer radius. This formula derives from integrating the area of circular cross-sections of the torus. Understanding this formula is crucial for solving problems related to the volume and dimensions of a torus.
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Differentiation
Differentiation is a fundamental concept in calculus that involves finding the rate at which a function changes at any given point. In this context, we need to differentiate the volume formula with respect to 'a' to find db/da, which represents how the outer radius 'b' changes as the inner radius 'a' changes while keeping the volume constant.
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Implicit Differentiation
Implicit differentiation is a technique used when dealing with equations where one variable is not explicitly solved for another. In this case, since the volume is constant (64π²), we can use implicit differentiation on the volume formula to relate the changes in 'a' and 'b'. This method allows us to find the derivative db/da without isolating 'b' in the equation.
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Related Practice
Textbook Question
13-26 Implicit differentiation Carry out the following steps.
a. Use implicit differentiation to find dy/dx.
³√x+³√y⁴ = 2;(1,1)
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Textbook Question
79–82. {Use of Tech} Visualizing tangent and normal lines <IMAGE>
a. Determine an equation of the tangent line and the normal line at the given point (x0, y0) on the following curves. (See instructions for Exercises 73–78.)
x⁴ = 2x²+2y²; (x0, y0)=(2, 2) (kampyle of Eudoxus)
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Textbook Question
A woman attached to a bungee cord jumps from a bridge that is 30 m above a river. Her height in meters above the river t seconds after the jump is y(t) = 15(1+e-t cos t), for t ≥ 0.
Determine her velocity at t = 1 and t = 3.
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Textbook Question
Comparing velocities Two stones are thrown vertically upward, each with an initial velocity of 48 ft/s at time t=0. One stone is thrown from the edge of a bridge that is 32 feet above the ground, and the other stone is thrown from ground level. The height above the ground of the stone thrown from the bridge after t seconds is f(t) = − 16t²+48t+32. and the height of the stone thrown from the ground after t seconds is g(t) = −16t²+48t.
a. Show that the stones reach their high points at the same time.
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Textbook Question
Derivatives and tangent lines
a. For the following functions and values of a, find f′(a).
f(x) = √3x; a= 12
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