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Chapter 6: Problem 70

Water in a bowl A hemispherical bowl of radius 8 inches is filled to a depth of \(h\) inches, where \(0 \leq h \leq 8 .\) Find the volume of water in the bowl as a function of \(h\). (Check the special cases \(h=0 \text { and } h=8 .)\)

Short Answer

Expert verified
Answer: The volume of the water in the bowl, as a function of the depth h, can be expressed as: \(V_\text{water} = \pi (8)^2 h - \frac{1}{3} \pi (64h - h^2)(8 - h)\).

Step by step solution

01

Determine the volume of the water as a cylinder

The volume of a cylinder is calculated using the formula \(V_\text{cylinder} = \pi r_\text{cylinder}^2 h\). In this case, the radius of the cylinder is equal to the radius of the bowl, 8 inches, and the height is equal to the depth of the water, h inches. Thus, we have: \(V_\text{cylinder} = \pi (8)^2 h\)
02

Determine the volume of the spherical cap above the water level

To find the volume of the spherical cap, we need to find the height and radius of the cap. We can use the Pythagorean theorem to relate the bowl's radius (8 inches) to the height and radius of the cap. Suppose the height of the cap is \(h_\text{cap}\) inches and its radius is \(r_\text{cap}\) inches. Then, \(h_\text{cap}^2 + r_\text{cap}^2 = 8^2\), and \(h_\text{cap} = \sqrt{8^2 - r_\text{cap}^2}\) Now, since the height of the cylindrical water is h inches, we can relate \(h_\text{cap}\) to h: \(h + h_\text{cap} = 8\) Substituting for \(h_\text{cap}\), we get: \(h + \sqrt{8^2 - r_\text{cap}^2} = 8\) Solving for \(r_\text{cap}\), we obtain: \(r_\text{cap}^2 = 8^2 - (8 - h)^2\) \(r_\text{cap}^2 = 64h - h^2\) The volume of the spherical cap is given by the formula \(V_\text{cap} = \frac{1}{3} \pi r_\text{cap}^2 h_\text{cap}\). Substituting the expressions for \(r_\text{cap}^2\) and \(h_\text{cap}\), we get: \(V_\text{cap} = \frac{1}{3} \pi (64h - h^2)(8 - h)\)
03

Determine the volume of the water in the bowl

The volume of the water in the bowl can be found by subtracting the volume of the spherical cap above the water level from the total volume of the cylindrical water. Therefore: \(V_\text{water} = V_\text{cylinder} - V_\text{cap}\) \(V_\text{water} = \pi (8)^2 h - \frac{1}{3} \pi (64h - h^2)(8 - h)\) This is our desired expression for the volume of the water in the bowl as a function of h.
04

Check the special cases \(h = 0\) and \(h = 8\)

When we have no water in the bowl (\(h = 0\)), the volume of the water, according to our function, should be 0: \(V_\text{water} = \pi (8)^2 (0) - \frac{1}{3} \pi (64(0) - (0)^2)(8 - 0)\) \(V_\text{water} = 0\) Thus, our function works for \(h = 0\). When the bowl is completely filled with water (\(h = 8\)), the volume, according to our function, should be equal to the volume of the hemisphere: \(V_\text{water} = \pi (8)^2 (8) - \frac{1}{3} \pi (64(8) - (8)^2)(8 - 8)\) \(V_\text{water} = \pi (8)^2 (8) - 0\) \(V_\text{water} = \frac{2}{3} \pi (8)^3\) This is equal to the volume of the hemisphere, so our function works for \(h = 8\) as well.

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Key Concepts

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

Spherical Cap Volume
Understanding the volume of a spherical cap is essential when dealing with partial volumes in spherical objects, such as a bowl partially filled with water. A spherical cap is the portion of a sphere that lies above or below a plane cutting through it.

In geometric terms, the volume of a spherical cap can be defined using the formula \(V_\text{cap} = \frac{1}{3} \pi h^2 (3R - h)\), where \(R\) is the radius of the sphere, and \(h\) is the height of the cap from the plane of intersection to the top of the cap. This formula encapsulates the relationship between the height and the curved surface area of the cap.

When calculating this volume in the context of a bowl filled with water, we can envision the empty part of the hemisphere as a spherical cap and use the formula above to determine its volume, thereby helping us find the volume occupied by the water by subtracting this from the full volume of the hemisphere.
Volume of a Cylinder
The concept of the volume of a cylinder is often used in tandem with other geometric formulas to solve complex problems. A cylinder is a three-dimensional shape with two parallel circular bases connected by a curved surface. The volume \(V\) of the cylinder is the amount of space contained within it and can be calculated with the familiar formula \(V = \pi r^2 h\), where \(r\) is the radius of the base circle and \(h\) is the height of the cylinder.

For the hemispherical bowl exercise, this calculation serves as the initial step to finding the volume of water by treating the water as if it were contained within a cylindrical volume before adjustments for the spherical shape are made.
Integrals in Geometry
The use of integrals in geometry allows us to calculate areas, volumes, and other properties involving curves and shapes. Integration is a concept from calculus that helps in finding the sum of infinite infinitesimal quantities. In terms of volume, integration can be applied to find the volume of irregular shapes by adding up the infinite number of infinitesimally thin disks or slices that make up the shape.

The process often involves establishing a function that provides the cross-sectional area of a figure at a given position and then integrating this function over a specific interval. For the spherical bowl, integration principles are applied in a more abstract manner through the derivation of the volume formulas for spheres and spherical caps.
Pythagorean Theorem
The Pythagorean theorem is one of the most recognized principles in geometry, expressing a fundamental relation in Euclidean geometry among the three sides of a right triangle. It states that the square of the length of the hypotenuse \(c\) is equal to the sum of the squares of the other two sides' lengths (\(a\) and \(b\)):

\(c^2 = a^2 + b^2\)

The theorem is not only useful in simple right-angle triangle calculations but also serves as a powerful tool in a wide array of geometric problems. In the context of finding the volume of water in a hemispherical bowl, the Pythagorean theorem helps determine the relationship between the radius of the bowl, the height of the water, and the radius of the spherical cap. By applying the theorem, we can solve for unknown quantities and eventually find the volume of water in the bowl.

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Most popular questions from this chapter

Emptying a cylindrical tank A cylindrical water tank has height \(8 \mathrm{m}\) and radius \(2 \mathrm{m}\) (see figure). a. If the tank is full of water, how much work is required to pump the water to the level of the top of the tank and out of the tank? b. Is it true that it takes half as much work to pump the water out of the tank when it is half full as when it is full? Explain.

Calculating work for different springs Calculate the work required to stretch the following springs \(1.25 \mathrm{m}\) from their equilibrium positions. Assume Hooke's law is obeyed. a. A spring that requires \(100 \mathrm{J}\) of work to be stretched \(0.5 \mathrm{m}\) from its equilibrium position b. A spring that requires a force of \(250 \mathrm{N}\) to be stretched \(0.5 \mathrm{m}\) from its equilibrium position

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. a. The mass of a thin wire is the length of the wire times its average density over its length. b. The work required to stretch a linear spring (that obeys Hooke's law) \(100 \mathrm{cm}\) from equilibrium is the same as the work required to compress it \(100 \mathrm{cm}\) from equilibrium. c. The work required to lift a 10 -kg object vertically \(10 \mathrm{m}\) is the same as the work required to lift a 20 -kg object vertically 5 m. d. The total force on a \(10-f t^{2}\) region on the (horizontal) floor of a pool is the same as the total force on a 10 -ft \(^{2}\) region on a (vertical) wall of the pool.

General slicing method Use the general slicing method to find the volume of the following solids. The solid whose base is the region bounded by the curve \(y=\sqrt{\cos x}\) and the \(x\) -axis on \([-\pi / 2, \pi / 2],\) and whose cross sections through the solid perpendicular to the \(x\) -axis are isosceles right triangles with a horizontal leg in the \(x y\) -plane and a vertical leg above the \(x\) -axis (IMAGE CAN'T COPY)

Lorenz curves and the Gini index A Lorenz curve is given by \(y=L(x),\) where \(0 \leq x \leq 1\) represents the lowest fraction of the population of a society in terms of wealth, and \(0 \leq y \leq 1\) represents the fraction of the total wealth that is owned by that fraction of the society. For example, the Lorenz curve in the figure shows that \(L(0.5)=0.2,\) which means that the lowest \(0.5(50 \%)\) of the society owns \(0.2(20 \%)\) of the wealth. (See Guided Project Distribution of Wealth for more on Lorenz curves.) a. A Lorenz curve \(y=L(x)\) is accompanicd by the line \(y=x\) called the line of perfect equality. Explain why this line is given this name. b. Explain why a Lorenz curve satisfies the conditions \(L(0)=0\) \(L(1)=1, L(x) \leq x,\) and \(L^{\prime}(x) \geq 0\) on [0,1] c. Graph the Lorenz curves \(L(x)=x^{p}\) corresponding to \(p=1.1\) \(1.5,2,3,\) and \(4 .\) Which value of \(p\) corresponds to the most equitable distribution of wealth (closest to the line of perfect equality)? Which value of \(p\) corresponds to the least equitable distribution of wealth? Explain. d. The information in the Lorenz curve is often summarized in a single measure called the Gini index, which is defined as follows. Let \(A\) be the area of the region between \(y=x\) and \(y=L(x)\) (see figure) and let \(B\) be the area of the region between \(y=L(x)\) and the \(x\) -axis. Then the Gini index is \(G=\frac{A}{A+B}\). Show that \(G=2 A=1-2 \int_{0}^{1} L(x) d x\) e. Compute the Gini index for the cases \(L(x)=x^{p}\) and \(p=1.1\) \(1.5,2,3,\) and 4 f. What is the smallest interval \([a, b]\) on which values of the Gini index lie for \(L(x)=x^{p}\) with \(p \geq 1 ?\) Which endpoints of \([a, b]\) correspond to the least and most equitable distribution of wealth? g. Consider the Lorenz curve described by \(L(x)=\frac{5 x^{2}}{6}+\frac{x}{6}\) Show that it satisfies the conditions \(L(0)=0, L(1)=1,\) and \(L^{\prime}(x) \geq 0\) on \([0,1] .\) Find the Gini index for this function.
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