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Chapter 7: Problem 12

Consider two solid figures, a sphere and cylinder, where each has radius \(r\). The volume of a sphere is \(V_{s}=\frac{4}{3} \pi r^{3}\) and the volume of a cylinder is \(V_{c}=\pi r^{2} h\). a. If \(h=1\), when is the volume of the sphere greater than the volume of the cylinder? b. What value of \(h\) would make the volumes the same? c. Complete this statement: "The volume of the cylinder with radius \(r\) is greater than the volume of a sphere of radius \(r,\) when \(h>\) ______ ;`

Short Answer

Expert verified
a. When \( r > \frac{3}{4} \)b. \( h = \frac{4}{3} r \)c. \( h > \frac{4}{3} r \).

Step by step solution

01

Write the given equations

The volume of the sphere is given by \[ V_{s} = \frac{4}{3} \pi r^{3} \] The volume of the cylinder is given by \[ V_{c} = \pi r^{2} h \]
02

Set up inequality for part (a)

For part (a), we need to determine when the volume of the sphere is greater than the volume of the cylinder if \( h = 1 \): \[ \frac{4}{3} \pi r^{3} > \pi r^{2} \times 1 \]
03

Solve the inequality

Divide both sides by \( \pi r^{2} \):\[ \frac{4}{3} r > 1 \] Simplify the inequality:\[ r > \frac{3}{4} \]
04

Solve for part (b)

Set the volume of the sphere equal to the volume of the cylinder to find the value of \( h \) that makes the volumes the same:\[ \frac{4}{3} \pi r^{3} = \pi r^{2} h \]Divide both sides by \( \pi r^{2} \):\[ \frac{4}{3} r = h \]Therefore, \( h = \frac{4}{3} r \)
05

Write the statement for part (c)

The volume of the cylinder with radius \( r \) is greater than the volume of a sphere with radius \( r \) when \( h > \frac{4}{3} r \)

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Key Concepts

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

sphere volume formula
A sphere is a perfectly round 3-dimensional shape, like a ball. The volume of a sphere is determined by the formula \( V_s = \frac{4}{3} \pi r^3 \). This formula tells us the amount of space inside the sphere. Here:
  • Vs is the volume of the sphere.
  • r is the radius of the sphere, which is the distance from the center to any point on its surface.
  • \( \pi \) is a mathematical constant approximately equal to 3.14159.
To calculate the volume, you cube the radius (multiply it by itself two more times), then multiply by \( \pi \), and finally by \( \frac{4}{3} \). This formula is derived through more advanced calculus concepts but is essential in geometry for understanding spherical shapes.
cylinder volume formula
A cylinder is a solid geometrical figure with straight parallel sides and a circular or oval cross-section. To find the volume of a cylinder, you use the formula \( V_c = \pi r^2 h \). This formula measures the space inside the cylinder. Here:
  • Vc is the volume of the cylinder.
  • r is the radius of the base, which is a circle.
  • h is the height of the cylinder, the distance between the two bases.
  • \( \pi \), as mentioned, is approximately 3.14159.
When you calculate, you first square the radius (multiply it by itself), then multiply by \( \pi \), and finally multiply by the height. This formula helps in all practical applications involving cylindrical objects, like cans or tubes.
inequalities in algebra
In algebra, an inequality is a relation that holds between two values when they are different. It shows that one value is either greater than, less than, greater than or equal to, or less than or equal to another value. The symbols used are:
  • \( > \) means greater than
  • \( < \) means less than
  • \( \geq \) means greater than or equal to
  • \( \leq \) means less than or equal to
For example, the inequality \( \frac{4}{3} r > 1 \) indicates that the radius \( r \) must be greater than \( \frac{3}{4} \) to meet the condition that the sphere's volume is more than the cylinder's volume (when \(h = 1\)). Solving inequalities often involves isolating the variable on one side of the inequality, similar to solving equations.
solving equations
Solving equations involves finding the value of the variables that make the equation true. The general goal is to isolate the variable on one side of the equation. Here鈥檚 how:
  • Move Terms: Shift terms from one side of the equation to the other to group all terms containing variables on one side and constants on the other.
  • Combine Like Terms: Simplify by combining like terms, which are terms with the same variable raised to the same power.
  • Isolate the Variable: Use basic operations (addition, subtraction, multiplication, and division) to isolate the variable.
For instance, to solve \( \frac{4}{3} r = h \) for \(h\), recognize that it鈥檚 already isolated. In our case, division was used in earlier steps to simplify the equation, showing how algebraic manipulation can help find relationships between different quantities.

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Most popular questions from this chapter

The frequency, \(F\) (the number of oscillations per unit of time), of an object of mass \(m\) attached to a spring is inversely proportional to the square root of \(m\). a. Write an equation describing the relationship. b. If a mass of \(0.25 \mathrm{~kg}\) attached to a spring makes three oscillations per second, find the constant of proportionality, c. Find the number of oscillations per second made by a mass of \(0.01 \mathrm{~kg}\) that is attached to the spring discussed in part (b).

Write a general formula to describe each variation. Use the information given to find the constant of proportionality. a. \(Q\) is directly proportional to both the cube root of \(t\) and the square of \(d\), and \(Q=18\) when \(t=8\) and \(d=3\). b. \(A\) is directly proportional to both \(h\) and the square of the radius, \(r\), and \(A=100 \pi\) when \(r=5\) and \(h=2\). c. \(V\) is directly proportional to \(B\) and \(h,\) and \(V=192\) when \(B=48\) and \(h=4\) d. \(T\) is directly proportional to both the square root of \(p\) and the square of \(u,\) and \(T=18\) when \(p=4\) and \(u=6\).

(Graphing program recommended.) In Exercise 15 we looked at how much a single load (a person's weight) could bend a plank downward by being placed at its midpoint. Now we look at what a continuous load, such as a solid row of books spread evenly along a shelf, can do. A long row of paperback fiction weighs about 10 pounds for each foot of the row. Typical hardbound books weigh about \(20 \mathrm{lb} / \mathrm{ft}\), and oversize hardbounds such as atlases, encyclopedias, and dictionaries weigh around \(36 \mathrm{lb} / \mathrm{ft}\). The following function is used to model the deflection \(D\) (in inches) of a \(1^{\prime \prime} \times 12^{\prime \prime}\) common pine board spanning a length of \(L\) feet carrying a continuous row of books: $$ D=\left(4.87 \cdot 10^{-4}\right) \cdot W \cdot L^{4} $$ where \(W\) is the weight per foot of the type of books along the shelf. This deflection model is quite good for deflections up to 1 inch; beyond that the fourth power causes the deflection value to increase very rapidly into unrealistic numbers. a. How much deflection does the formula predict for a shelf span of 30 inches with oversize books? Would you recommend a stronger, thicker shelf? b. Plot \(D_{\text {hardbound }}, D_{\text {paperback }},\) and \(D_{\text {oversize }}\) on the same graph. Put \(L\) on the horizontal axis with values up to 4 feet. c. For each kind of book identify what length, \(L,\) will cause a deflection of 0.5 inch in a \(1^{\prime \prime} \times 12^{\prime \prime}\) pine shelf.

(Graphing program recommended.) Graph \(y=x^{4}\) and \(y=4^{x}\) on the same grid. a. For positive values of \(x\), where do your graphs intersect? Do they intersect more than once? b. For positive values of \(x\), describe what happens to the right and left of any intersection points. You may need to change the scales on the axes or change the windows on a graphing calculator in order to see what is happening. c. Which eventually dominates, \(y=x^{4}\) or \(y=4^{x} ?\)

The volume, \(V,\) of a cylindrical can is \(V=\pi r^{2} h\) and the total surface area, \(S\), of the can is $$ \begin{aligned} S &=\text { area of curved surface }+2 \cdot \text { (area of base) } \\ &=2 \pi r h+2 \pi r^{2} \end{aligned} $$ where \(r\) is the radius of the base and \(h\) is the height. a. Assume the height is three times the radius. Write the volume and the surface area as functions of the radius. b. As \(r\) increases, which grows faster, the volume or the surface area? Explain.
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