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Chapter 0: Problem 27

\(V(r)=\frac{4}{3} \pi r^{3}\) (a) \(V(3)\) (b) \(V\left(\frac{3}{2}\right)\) (c) \(V(2 r)\)

Short Answer

Expert verified
The volumes calculated for each case are: (a) \(V(3) = 36 \pi\) cubic units, (b) \(V(3/2) = \frac{9}{2} \pi\) cubic units, (c) \(V(2r) = 32 \pi r^{3}\) cubic units.

Step by step solution

01

Substitute r = 3 into the formula

Substitute r = 3 into \(V(r) = \frac{4}{3} \pi r^{3}\) which gives \(V(3) = \frac{4}{3} \pi (3)^{3}\). Perform the calculation to get the volume.
02

Substitute r = 3/2 into the formula

Substitute r = 3/2 into \(V(r) = \frac{4}{3} \pi r^{3}\) which gives \(V(3/2) = \frac{4}{3} \pi (3/2)^{3}\). Perform the calculation to get the volume.
03

Substitute r = 2r into the formula

Substitute r = 2r into \(V(r) = \frac{4}{3} \pi r^{3}\) which gives \(V(2r) = \frac{4}{3} \pi (2r)^{3}\). Notice that this calculation involves an extra step of simplifying \( (2r)^{3}\) to \(8r^{3}\) before proceeding with the calculation.

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Key Concepts

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

Substitution in Formulas
When working with formulas, especially in geometry, substitution is a key technique that involves replacing variables with specific values. This process allows you to find particular numerical results from a general equation.
For instance, in the given exercise, the formula \(V(r)=\frac{4}{3} \pi r^{3}\) is used to calculate the volume of a sphere. Here, \(r\) represents the radius, which is a variable placeholder that can be replaced with different values to perform specific calculations.
  • In part (a), we substitute \(r = 3\).
  • In part (b), \(r = \frac{3}{2}\) is used.
  • In part (c), the substitution \(r = 2r\) indicates calculating a volume with double the original radius.
Substitution helps in determining how the volume changes as the radius changes. It is an essential practice in applying mathematical models to real-world scenarios.
Calculating Volume
To calculate the volume of a sphere, we use the formula \(V(r)=\frac{4}{3} \pi r^{3}\). This is a standard formula used in geometry, which describes how to find the amount of space inside a sphere.
The process involves substituting the given value of the radius \(r\) into this equation and then simplifying the expression. Here's a brief breakdown:
  • First, substitute the given radius into the formula. For example, for \(r = 3\), the formula becomes \(V(3) = \frac{4}{3} \pi (3)^{3}\).
  • Next, compute the cube of the radius. For \((3)^{3}\), this equals 27. Similarly, \((\frac{3}{2})^{3}\) equals \(\frac{27}{8}\) and \((2r)^{3}\) simplifies to \(8r^{3}\).
  • Finally, multiply by \(\frac{4}{3}\pi\) to find the exact volume.
Using these steps, you calculate the volume effectively, ensuring that your mathematical operations are correct.
Radius in Geometry
Understanding the concept of radius is crucial in geometry. It is defined as the distance from the center of a circle or sphere to its surface. The radius is crucial in calculating various properties of geometric shapes, particularly those of circular forms.
In the context of a sphere, the radius \(r\) directly affects the volume of the sphere because the volume formula \(V(r)=\frac{4}{3} \pi r^{3}\) depends on \(r\) to the power of three.
  • The radius serves as an initial step in setting up equations involving circles and spheres, providing a key input in calculations.
  • Understanding how changes in the radius impact the overall geometry can provide insight into real-world applications, such as designing spherical objects like balls or domes.
  • Notably, when the radius is doubled, like in part (c) of the exercise, the overall volume becomes eight times larger due to the cubic relationship: \((2r)^{3} = 8r^{3}\).
The geometric properties connected to the radius show how integral this measure is to understanding space and form in mathematics.

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Most popular questions from this chapter

Prescription Drugs The amounts \(d\) (in billions of dollars) spent on prescription drugs in the United States from 1991 to 2002 (see figure) can be approximated by the model $$ d(t)= \begin{cases}5.0 t+37, & 1 \leq t \leq 7 \\ 18.7 t-64, & 8 \leq t \leq 12\end{cases} $$ where \(t\) represents the year, with \(t=1\) corresponding to 1991. Use this model to find the amount spent on prescription drugs in each year from 1991 to 2002 . (Source: U.S. Centers for Medicare \& Medicaid Services)

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Average Price The average prices \(p\) (in thousands of dollars) of a new mobile home in the United States from 1990 to 2002 (see figure) can be approximated by the model $$ p(t)= \begin{cases}0.182 t^{2}+0.57 t+27.3, & 0 \leq t \leq 7 \\ 2.50 t+21.3, & 8 \leq t \leq 12\end{cases} $$ where \(t\) represents the year, with \(t=0\) corresponding to 1990. Use this model to find the average price of a mobile home in each year from 1990 to 2002 . (Source: U.S. Census Bureau)
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