Problem 33
Question
The radius, \(r\), in \(\mathrm{cm}\), of a sphere of volume \(V \mathrm{~cm}^{3}\) is approximately \(r=0.620 \sqrt[3]{V}\). (a) Graph the radius function, \(r,\) for volumes from 0 to \(40 \mathrm{~cm}^{3}\). (b) Use your graph to estimate the volume of a sphere of radius \(2 \mathrm{~cm} .\)
Step-by-Step Solution
Verified Answer
Answer:
To find the radius of a sphere when its volume is 20 cm^3, use the given formula \(r = 0.620 \sqrt[3]{V}\). Plug in the volume (V = 20 cm^3) into the formula and calculate the corresponding radius value. To estimate this using the graph, locate the point on the graph where the volume is 20 cm^3 and read the corresponding radius value from the graph.
1Step 1: Create a set of volume values and calculate the corresponding radius values
First, create volume values, \(V\), between 0 and 40 cm^3 at regular intervals (for example, every 5 cm^3). Next, calculate the respective radius, \(r\), values using the given formula \(r = 0.620 \sqrt[3]{V}\).
2Step 2: Plot the radius function
On a graph paper or using graphing software, plot the calculated points from Step 1. Then, connect the points with a smooth curve to represent the function \(r = 0.620 \sqrt[3]{V}\).
3Step 3: Estimate the volume of a sphere with a radius of 2 cm
Using the plotted graph, identify the point where the radius, \(r\), is 2 cm and read the corresponding volume, \(V\), value from the graph.
Key Concepts
Graphing FunctionsRadius-Volume RelationshipEstimating Volume from Graphs
Graphing Functions
Graphing functions is a powerful mathematical technique that helps us visualize relationships between different variables. In this exercise, you are asked to create a graph for the sphere's radius function. The function in question is given by the relationship \(r = 0.620 \sqrt[3]{V}\), where \(r\) is the radius and \(V\) is the volume of the sphere.
To graph this function, you'll need to determine various points by choosing a series of volume values within the specified range. Calculating the corresponding radius for each of these volumes can be done using the provided formula. A handy approach is to pick intervals at which you wish to calculate these values, such as every 5 cm³. This systematic approach ensures you cover an appropriate range and can lead to a more accurate shape of the function when plotted.
When plotting these points, either on graph paper or using digital tools, it's crucial to maintain consistent scales on both the horizontal (volume) and vertical (radius) axes. To complete the graph, these points should be connected smoothly, reflecting the continuous nature of the mathematical function.
To graph this function, you'll need to determine various points by choosing a series of volume values within the specified range. Calculating the corresponding radius for each of these volumes can be done using the provided formula. A handy approach is to pick intervals at which you wish to calculate these values, such as every 5 cm³. This systematic approach ensures you cover an appropriate range and can lead to a more accurate shape of the function when plotted.
When plotting these points, either on graph paper or using digital tools, it's crucial to maintain consistent scales on both the horizontal (volume) and vertical (radius) axes. To complete the graph, these points should be connected smoothly, reflecting the continuous nature of the mathematical function.
Radius-Volume Relationship
The relationship between radius and volume in a sphere is illustrated by the formula \(r = 0.620 \sqrt[3]{V}\). This expression indicates how radius and volume are connected in a non-linear fashion. The cube root implies that changes in the volume result in less-than-proportional changes in the radius. As a sphere's volume increases, the radius increases but at a diminishing rate due to the cube root effect.
Breaking down this relationship:
Breaking down this relationship:
- Cubic Inversion: The radius grows as the cube root of the volume, denoted by \(\sqrt[3]{V}\), showcasing a need for larger volumes to significantly impact radius size.
- Proportional Constant: The 0.620 simplifies the connection, adjusting the cube root relationship to deliver a radius in centimeters directly from volume measurements.
Estimating Volume from Graphs
Estimating values from a graph involves identifying how certain variables correlate based on visual representations. After plotting the radius against volume for our specific function, you can use this graph to find unknown values, such as determining the volume for a known radius.
To estimate the volume of a sphere with a known radius, say 2 cm in this scenario, one would locate the radius on the vertical axis of the graph and follow horizontally until it intersects with the curve. Dropping vertically from this point of intersection to meet the horizontal axis provides the approximate volume value for the sphere corresponding to a radius of 2 cm.
This method of estimation is particularly useful when using graphs in practical scenarios. By visually capturing trends, it becomes straightforward to evaluate or predict one variable when others are known, leveraging the power of visual data intuition and mathematical foundations.
To estimate the volume of a sphere with a known radius, say 2 cm in this scenario, one would locate the radius on the vertical axis of the graph and follow horizontally until it intersects with the curve. Dropping vertically from this point of intersection to meet the horizontal axis provides the approximate volume value for the sphere corresponding to a radius of 2 cm.
This method of estimation is particularly useful when using graphs in practical scenarios. By visually capturing trends, it becomes straightforward to evaluate or predict one variable when others are known, leveraging the power of visual data intuition and mathematical foundations.
Other exercises in this chapter
Problem 32
The surface area of a mammal is given by \(f(M)=\) \(k M^{2 / 3},\) where \(M\) is the body mass, and the constant of proportionality \(k\) is a positive number
View solution Problem 32
\(a\) and \(b\) are positive numbers and \(a>b\). Which is larger, \(f(a)\) or \(f(b)\) ? $$ f(t)=\frac{\sqrt{5}}{t^{7}} $$
View solution Problem 33
\(a\) and \(b\) are positive numbers and \(a>b\). Which is larger, \(f(a)\) or \(f(b)\) ? $$ f(r)=\sqrt{\frac{12}{r}} $$
View solution Problem 34
\(a\) and \(b\) are positive numbers and \(a>b\). Which is larger, \(f(a)\) or \(f(b)\) ? $$ f(t)=-3 t^{2}\left(-2 t^{3}\right)^{5} $$
View solution