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Chapter 12: Problem 70

The volume of the cap of a sphere of radius \(r\) and thickness \(h\) is \(V=\frac{\pi}{3} h^{2}(3 r-h),\) for \(0 \leq h \leq 2 r\). a. Compute the partial derivatives \(V_{h}\) and \(V_{r}\). b. For a sphere of any radius, is the rate of change of volume with respect to \(r\) greater when \(h=0.2 r\) or when \(h=0.8 r ?\) c. For a sphere of any radius, for what value of \(h\) is the rate of change of volume with respect to \(r\) equal to \(1 ?\) d. For a fixed radius \(r,\) for what value of \(h(0 \leq h \leq 2 r)\) is the rate of change of volume with respect to \(h\) the greatest?

Short Answer

Expert verified
Answer: The rate of change of volume with respect to r is greater when h = 0.8r. Question: Determine the value of h for which the rate of change of volume with respect to r is equal to 1. Answer: The value of h for which the rate of change of volume with respect to r is equal to 1 is h = 鈭�(1/(3蟺)). Question: Find the value of h for which the rate of change of volume with respect to h is the greatest, when r is fixed. Answer: The value of h that maximizes the rate of change of volume with respect to h when r is fixed is h = 3/2 (r being fixed and positive).

Step by step solution

01

Compute partial derivatives of V with respect to r and h

The given formula for the volume of the cap of a sphere is: \(V = \frac{\pi}{3}h^2(3r-h)\) To compute the partial derivative of \(V\) with respect to \(h\), we will treat \(r\) as a constant and differentiate the formula with respect to \(h\). Similarly, to compute the partial derivative of \(V\) with respect to \(r\), we will treat \(h\) as a constant and differentiate the formula with respect to \(r\). - Partial derivative with respect to \(h\): \(V_h = \frac{dV}{dh} = \frac{\pi}{3}(2h(3r-h) - h^2)\) - Partial derivative with respect to \(r\): \(V_r = \frac{dV}{dr} = \frac{\pi}{3}(h^2(3))\)
02

Analyze the rate of change of volume with respect to r for given values of h

We need to compare the rate of change of volume with respect to \(r\) when \(h = 0.2r\) and \(h = 0.8r\). We will substitute these values into the expression for \(V_r\) and compare the resulting rates: - When \(h = 0.2r\): \(V_r = \frac{\pi}{3}((0.2r)^2(3)) = \frac{4\pi r^2}{15}\) - When \(h = 0.8r\): \(V_r = \frac{\pi}{3}((0.8r)^2(3)) = \frac{16\pi r^2}{15}\) Comparing the two values, we can see that the rate of change of volume with respect to \(r\) is greater when \(h = 0.8r\).
03

Determine the value of h for which the rate of change of volume with respect to r is equal to 1

We will set the expression for \(V_r\) equal to \(1\) and solve for \(h\): \(\frac{\pi}{3}(h^2(3)) = 1\) \(h^2 = \frac{3}{9\pi}\) \(h = \sqrt{\frac{1}{3\pi}}\)
04

Find the value of h for which the rate of change of volume with respect to h is the greatest, when r is fixed

To find the value of \(h\) that maximizes the rate of change of volume with respect to \(h\), we will find the critical points of \(V_h\). This means we will set the second partial derivative of \(V\) with respect to \(h\) equal to \(0\) and solve for \(h\): \(V_{hh} = \frac{d^2V}{dh^2} = \frac{\pi}{3}(6-4h)\) Setting \(V_{hh}\) equal to 0 to find the critical points: \(\frac{\pi}{3}(6-4h) = 0\) \(6-4h = 0\) \(h = \frac{3}{2}\) Since \(0 \leq h \leq 2r\), we have that \(h=\frac{3}{2}\) maximizes the rate of change of volume with respect to \(h\) when \(r\) is fixed.

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Key Concepts

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

Rate of Change
The concept of rate of change is central in understanding how a quantity changes concerning another. In this exercise, we deal with partial derivatives, which allow us to examine how a function changes as each of its variables changes, keeping other variables constant. In our case, the volume of the sphere's cap changes based on two variables: its radius \( r \) and its thickness \( h \).

Partial derivatives, like \( V_h \) and \( V_r \), are derivatives of the volume function \( V \) with respect to \( h \) and \( r \) respectively. They help us understand the sensitivity of the volume to changes in either \( h \) or \( r \).

The exercise guides us to
  • calculate partial derivatives,
  • compare changes at different thickness proportions,
  • identify specific conditions (like when these rates equal a particular value),
  • determine where these rates are maximized.
This understanding is vital for solving practical problems involving geometrical changes in structures.
Sphere Volume
The volume of a sphere, or portions of a sphere, like the cap specified in this exercise, is a classic example of three-dimensional geometry. The formula provided, \( V=\frac{\pi}{3} h^2(3 r-h) \), describes how the cap's volume varies with the sphere's radius \( r \) and its cap height or thickness \( h \).

Understanding this formula requires breaking down its components:
  • \( \frac{\pi}{3} \) is a scaling factor common in spherical volume calculations.
  • \( h^2 \) indicates how changes in height have a squared effect on volume.
  • \( (3r-h) \) shows the linear dependency on the sphere radius and cap height.
These components interact to determine how any alteration in \( r \) or \( h \) influences the cap's total volume. Recognizing these interactions is essential for predicting and managing changes in real-world applications, such as volume maintenance in engineering or manufacturing processes.
Critical Points
Critical points are values in mathematical functions where derivatives (either first or second) are zero, potentially indicating maxima, minima, or points of inflection. In this exercise, finding critical points helps us identify thickness levels where the volume's sensitivity to change, concerning the cap's thickness, is highest.

To identify critical points for \( V_h \), we first compute the second partial derivative \( V_{hh} \). Setting \( V_{hh} \) to zero helps us find these critical values for \( h \).

Once identified, critical points guide us to:
  • Maximize efficiency by highlighting the most significant change rates,
  • Optimize engineering applications,
  • Understand the most impactful adjustments in scenarios involving spherical structures or similar geometries.
By strategically leveraging critical point data, we can optimize designs and processes for peak performance and safety.

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Most popular questions from this chapter

Let \(R\) be a closed bounded region in \(\mathbb{R}^{2}\) and let \(f(x, y)=a x+b y+c,\) where \(a, b\) and \(c\) are real numbers, with \(a\) and \(b\) not both zero. Give a geometrical argument explaining why the absolute maximum and minimum values of \(f\) over \(R\) occur on the boundaries of \(R\).

Identify and briefly describe the surfaces defined by the following equations. $$y=4 z^{2}-x^{2}$$

Suppose \(n\) houses are located at the distinct points \(\left(x_{1}, y_{1}\right),\left(x_{2}, y_{2}\right), \ldots,\left(x_{n}, y_{n}\right) .\) A power substation must be located at a point such that the sum of the squares of the distances between the houses and the substation is minimized. a. Find the optimal location of the substation in the case that \(n=3\) and the houses are located at \((0,0),(2,0),\) and (1,1) b. Find the optimal location of the substation in the case that \(n=3\) and the houses are located at distinct points \(\left(x_{1}, y_{1}\right)\) \(\left(x_{2}, y_{2}\right),\) and \(\left(x_{3}, y_{3}\right)\) c. Find the optimal location of the substation in the general case of \(n\) houses located at distinct points \(\left(x_{1}, y_{1}\right),\left(x_{2}, y_{2}\right), \ldots\) \(\left(x_{n}, y_{n}\right)\) d. You might argue that the locations found in parts (a), (b), and (c) are not optimal because they result from minimizing the sum of the squares of the distances, not the sum of the distances themselves. Use the locations in part (a) and write the function that gives the sum of the distances. Note that minimizing this function is much more difficult than in part (a).

Consider the ellipse \(x^{2}+4 y^{2}=1\) in the \(x y\) -plane. a. If this ellipse is revolved about the \(x\) -axis, what is the equation of the resulting ellipsoid? b. If this ellipse is revolved about the \(y\) -axis, what is the equation of the resulting ellipsoid?

Let \(f(x, y)=0\) define \(y\) as a twice differentiable function of \(x\) a. Show that \(y^{\prime \prime}(x)=\frac{f_{x x} f_{y}^{2}-2 f_{x} f_{y} f_{x y}+f_{y y} f_{x}^{2}}{f_{y}^{3}}\) b. Verify part (a) using the function \(f(x, y)=x y-1\)
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