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Chapter 4: Problem 39

Volume The change in the volume \(V=\pi r^{2} h\) of a right circular cylinder when the radius changes from \(a\) to \(a+d r\) and the height does not change

Short Answer

Expert verified
The change in volume when the radius of the right circular cylinder changes from \(a\) to \(a+dr\) is \(2\pi a h dr\).

Step by step solution

01

Understanding the volume formula of a cylinder

The given formula \(V=\pi r^{2} h\) tells us that the volume of a cylinder is equal to the area of its base (which is a circle with radius \(r\), so its area is \(\pi r^{2}\)) multiplied by its height \(h\).
02

Differentiating the volume formula

We want to find the change in volume with respect to a small change in radius. This requires differentiating the volume formula with respect to \(r\). Using the power rule that the derivative of \(r^{2}\) is \(2r\), we get the derivative of the volume formula as \(dV = 2\pi r h dr\).
03

Substituting the values of radius and change in radius

To find the actual change in volume when the radius changes from \(a\) to \(a+dr\), we substitute \(r=a\) and \(dr=dr\) into the equation from Step 2: \(dV = 2\pi a h dr\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Differential Calculus
Differential calculus is a subfield of calculus concerned with the notion of a derivative, which represents an instantaneous rate of change. This is particularly useful when analyzing how a small change in one quantity leads to a change in another. In geometry and physics, we often use differential calculus to determine how objects alter their shape or velocity.

With the example of a cylinder's volume changing, differential calculus allows us to calculate the effect of a slight alteration in one dimension, say the radius, while keeping others constant. By understanding the principles of differential calculus, we can predict and understand how real-world systems behave under small perturbations.
Volume of a Cylinder
The volume of a cylinder is a measurement that tells us how much three-dimensional space it occupies. The formula for the volume is given by the product of the base area and the height of the cylinder, expressed as \( V = \text{pi} r^{2} h \).

Where \( \text{pi} \) is a mathematical constant (approximately 3.14159), \( r \) is the radius of the cylinder's circular base, and \( h \) is its height. Understanding this formula is crucial for solving problems related to the storage, displacement, or manufacturing of cylindrical objects.
Power Rule
The power rule is a basic and extremely useful principle in differential calculus for finding derivatives. It states that if you have a function of the form \( f(x) = x^n \), where \( n \) is any real number, then the derivative of that function with respect to \( x \) is \( f'(x) = nx^{n-1} \).

For instance, the derivative of \( r^{2} \) with respect to \( r \) is \( 2r \), since here \( n = 2 \). The power rule simplifies the process of differentiation greatly, making it easier to solve problems involving rates of change in variables raised to a power, such as the radius in the volume of a cylinder.
Derivatives in Geometry
Derivatives play a significant role in geometry, especially when determining how changes in dimensions affect various geometric properties such as volume and surface area. In the context of a cylinder, if we want to know how a small change in the radius affects the overall volume, we would take the derivative of the volume formula with respect to the radius.

This application of derivatives helps architects, engineers, and designers to calculate necessary materials, understand structural stability, and optimize shapes for different functions. By mastering derivatives in the realm of geometry, students can gain valuable insights into the relationships between different geometric dimensions.

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Most popular questions from this chapter

In Exercises 23 and \(24,\) a particle is moving along the curve \(y=f(x) .\) \(y=f(x)=\frac{10}{1+x^{2}}\) If \( \)d x / d t=3 \mathrm{cm} / \mathrm{sec}, \text { find } d y / d t \(d x / d t=3 \mathrm{cm} / \mathrm{sec},\) find \(d y / d t\) at the point where $$x=-2 \text { . } \quad \text { (b) } x=0 . \quad \text { (c) } x=20$$

Draining Hemispherical Reservoir Water is flowing at the rate of 6 \(\mathrm{m}^{3} / \mathrm{min}\) from a reservoir shaped like a hemispherical bowl of radius \(13 \mathrm{m},\) shown here in profile. Answer the following questions given that the volume of water in a hemispherical bowl of radius \(R\) is \(V=(\pi / 3) y^{2}(3 R-y)\) when the water is \(y\) units deep. (a) At what rate is the water level changing when the water is 8 m deep? (b) What is the radius \(r\) of the water's surface when the water is \(y\) m deep? (c) At what rate is the radius \(r\) changing when the water is 8 \(\mathrm{m}\) deep?

The Linearization is the Best Linear Approximation Suppose that \(y=f(x)\) is differentiable at \(x=a\) and that \(g(x)=m(x-a)+c(m\) and \(c\) constants). If the error \(E(x)=f(x)-g(x)\) were small enough near \(x=a,\) we might think of using \(g\) as a linear approximation of \(f\) instead of the linearization \(L(x)=f(a)+f^{\prime}(a)(x-a) .\) Show that if we impose on \(g\) the conditions i. \(E(a)=0\) ii. \(\lim _{x \rightarrow a} \frac{E(x)}{x-a}=0\) then \(g(x)=f(a)+f^{\prime}(a)(x-a) .\) Thus, the linearization gives the only linear approximation whose error is both zero at \(x=a\) and negligible in comparison with \((x-a)\) .

Draining Conical Reservoir Water is flowing at the rate of 50 \(\mathrm{m}^{3} / \mathrm{min}\) from a concrete conical reservoir (vertex down) of base radius 45 \(\mathrm{m}\) and height 6 \(\mathrm{m} .\) (a) How fast is the water level falling when the water is 5 \(\mathrm{m}\) deep? (b) How fast is the radius of the water's surface changing at that moment? Give your answer in \(\mathrm{cm} / \mathrm{min.}\)

Speed Trap A highway patrol airplane flies 3 mi above a level, straight road at a constant rate of 120 mph. The pilot sees an oncoming car and with radar determines that at the instant the line-of-sight distance from plane to car is 5 mi the linstant the distance is decreasing at the rate of 160 \(\mathrm{mph}\) . Find the car's speed along the highway.
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