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Chapter 1: Problem 38

The height of a right circular cone is one third of the diameter of the base. (a) Express its volume as a function of its height, \(h\). (b) Express its volume as a function of \(r\), the radius of its base.

Short Answer

Expert verified
(a) The volume of the cone as a function of \( h \) is \( V = \frac{3}{4}\pi h^3 \). (b) The volume of the cone as a function of \( r \) is \( V = \frac{2}{9}\pi r^3 \)

Step by step solution

01

- Express the Diameter in Terms of \( h \)

Given the height of a right circular cone is one third of the diameter of the base, we write the diameter \( d \) in terms of \( h \). So, \( d = 3h \)
02

- Express Volume as a Function of \( h \)

Since the diameter \( d \) is twice the radius, we substitute \( r \) with \( \frac{d}{2} = \frac{3h}{2} \) in the volume formula, obtaining \( V = \frac{1}{3} \pi \left(\frac{3h}{2}\right)^2h \). After simplification, the formula becomes \( V = \frac{9}{12}\pi h^3 = \frac{3}{4}\pi h^3 \)
03

- Express Volume as a Function of \( r \)

We substitute \( h \) with \( \frac{2r}{3} \) in the volume formula, producing \( V = \frac{1}{3} \pi r^2(\frac{2r}{3}) \). After simplification, the formula becomes \( V = \frac{2}{9}\pi r^3 \)

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Key Concepts

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

Height and Diameter Relationship
In the context of geometry, understanding the relationship between the height and diameter of a cone is crucial. When we talk about a right circular cone, the height is perpendicular from the base to its apex. In this scenario, the given height is one third of the diameter of the base. This means that if you know the diameter, you can quickly find the height and vice versa, by utilizing simple multiplication or division.
The formula given in the exercise is:
  • For diameter, if height is given: \(d = 3h\)
  • For height, if diameter is given: \(h = \frac{d}{3}\)
This relationship helps in setting up the equations correctly when calculating the cone's volume.
Volume Formula
The volume of a cone is the amount of space enclosed by it, and it can be calculated using its height and radius of the base. The standard volume formula for a cone is given by:
  • \( V = \frac{1}{3} \pi r^2 h \)
This formula uses three key elements:
  • \(\pi\): A constant, approximately 3.14159, that relates to the circle's circumference and area.
  • \(r^2\): The square of the radius of the base.
  • \(h\): The height of the cone.
Plug these values into the formula to find the cone's volume. Remember, finding the volume means calculating how much space is inside the cone, which is usually expressed in cubic units.
Right Circular Cone
A right circular cone is defined by having a circular base and a vertex or apex that is aligned above the center of the base. This makes the perpendicular distance from the base to the apex the height of the cone. This shape comes into play when dealing with many mathematical problems and real-world applications such as designing funnels or ice cream cones.
Key characteristics include:
  • The base is circular
  • The height is perpendicular to the base
  • The vertex is directly above the base's center
Understanding these properties aids in deriving important formulas and equations related to cones, ensuring accurate mathematical solutions.
Radius of a Circle
The radius is an important metric in understanding and calculating the dimensions of a cone鈥檚 base. It is defined as the distance from the center of a circle to any point on its circumference. Knowing the radius allows calculation of other crucial factors such as the diameter and the area of the circle.
In the case of cones, it also plays a significant part in determining the volume. The radius can often be denoted as \(r\) in equations. Here鈥檚 a quick breakdown:
  • The diameter \(d\) is twice as long as the radius: \(d = 2r\)
  • The area of the circle is expressed as \( \pi r^2 \)
In volume calculations, the square of the radius is very significant because it directly affects the overall volume of the cone, making understanding and correctly calculating the radius vital for accurate results.

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Most popular questions from this chapter

Let \(C\) be a circle of radius 1 and let \(A(n)\) be the area of a regular \(n\) -gon inscribed in the circle. For instance, \(A(3)\) is the area of an equilateral triangle inscribed in circle \(C, A(4)\) is the area of a square inscribed in circle \(C\), and \(A(5)\) is the area of a regular pentagon inscribed in circle \(C\). (A polygon inscribed in a circle has all its vertices lying on the circle. A regular polygon is a polygon whose sides are all of equal length and whose angles are all of equal measure.) (a) Find \(A(4)\). (b) Is \(A(n)\) a function? If it is, answer the questions that follow. (c) What is the natural domain of \(A(n)\) ? (d) As \(n\) increases, do you think that \(A(n)\) increases, or decreases? This is hard to justify rigorously, but what does your intuition tell you? (e) Will \(A(n)\) increase without bound as \(n\) increases, or is there a lid above which the values of \(A(n)\) will never go? If there is such a lid (called an upper bound) give one. What is the smallest lid possible? Rigorous justification is not requested.

A closed rectangular box has a square base. Let \(s\) denote the length of the sides of the base and let \(h\) denote the height of the box, \(s\) and \(h\) in inches. (a) Express the volume of the box in terms of \(s\) and \(h\). (b) Express the surface area of the box in terms of \(s\) and \(h\). (c) If the volume of the box is 120 cubic inches, express the surface area of the box as a function of \(s\).

During road construction gravel is being poured onto the ground from the top of a tall truck. The gravel falls into a conical pile whose height is always equal to half of its radius. Express the amount of gravel in the pile as a function of its (a) height. (b) radius.

Two bears, Bruno and Lollipop, discover a patch of huckleberries one morning. The patch covers an area of \(A\) acres and there are \(X\) bushels of huckleberries per acre. Bruno eats \(B\) bushels of huckleberries per hour; Lollipop can devour \(L\) bushels of huckleberries in \(C\) hours. Express your answers to parts (a) and (b) in terms of any or all of the constants \(A, X, B, L\), and \(C .\) (a) Express the number of bushels of huckleberries the two bears eat as a function of \(t\), the number of hours they have been eating. (b) In \(t\) hours, how many acres of huckleberries can the two bears together finish off? (c) Assuming that after \(T\) hours the bears have not yet finished the berry patch, how many hours longer does it take them to finish all the huckleberries in the patch? Express your answers in terms of any or all of the constants \(A, X, B, L, C\), and \(T\). If you are having difficulty, use this time-tested technique: Give the quantity you are looking for a name. (Avoid the letters already standing for something else.)

Assume that \(f\) is a function with domain \((-\infty, \infty)\). Which of the following statements is true for every such function \(f\) and all \(p, w\), and \(z\) in the domain of \(f ?\) If a statement is not true for every function, find a function for which it is false. (Hint: The constant functions are good functions to use as a first check.) (a) \(f(2) f(3)=f(6)\) (b) \(f(p)+f(p)=2 f(p)\) (c) \(f(4+5)=f(4)+f(5)\) (d) \(f(w) f(w)=[f(w)]^{2}\) (e) \(f(z) f(z)=f\left(z^{2}\right)\) (f) \(\frac{f(x)}{5[f(x)]^{2}}=\frac{1}{5 f(x)}\), where \(f(x) \neq 0\) (g) \(f(1) f(7)=f(7)\)
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