The formula for the volume of a sphere is V=(4)/(3) πr^3, where r represents the radius of the s

step 1 In first part, we have to write a function DR that gives the volume gained when the radius of th

The formula for the volume of a sphere is V=(4)/(3) πr^3,...

The formula for the volume of a sphere is $V=\frac{4}{3} \pi r^{3},$ where $r$ represents the radius of the sphere. (a) Write a function $D(r)$ that gives the volume gained when the radius of a sphere of $r$ inches is increased by 3 inches. (b) Graph $y=D(r)$ found in part (a), using $x$ for $r,$ in the window $[0,10]$ by $[0,1500]$ (c) Use your calculator to graphically find the amount of volume gained when a sphere with a 4-inch radius is increased to a 7 -inch radius. (d) Verify your result in part (c) analytically.

The formula for the volume of a sphere is $V=\frac{4}{3} \pi r^{3},$ where $r$ represents the radius of the sphere.
(a) Write a function $D(r)$ that gives the volume gained when the radius of a sphere of $r$ inches is increased by 3 inches.
(b) Graph $y=D(r)$ found in part (a), using $x$ for $r,$ in the window $[0,10]$ by $[0,1500]$
(c) Use your calculator to graphically find the amount of volume gained when a sphere with a 4-inch radius is increased to a 7 -inch radius.
(d) Verify your result in part (c) analytically.

Labs

Want to see this concept in action?

NEW

Explore this concept interactively to see how it behaves as you change inputs.

View Labs

Key Concepts

Analytical Verification

This concept emphasizes confirming the results obtained from graphical or numerical methods by using algebraic or calculus-based approaches. It ensures that the computations and reasoning behind the modeled function are precise and accurate.

Volume of a Sphere

This key concept involves understanding and using the geometric formula V = (4/3)?r³ to compute the space enclosed by a sphere. It forms the basis for further calculations involving changes in volume as the sphere's dimensions are altered.

Function Modeling

This concept pertains to defining functions to represent real-world scenarios mathematically. In this context, creating a function that expresses the volume gain as the radius increases allows for systematic analysis and evaluation of the relationship between the sphere’s size and its volume.

Graphical Representation

Graphing a function involves plotting the relationship between variables on a coordinate system to visually analyze behavior over a specified domain and range. It provides insight into how the volume gain varies with the sphere's radius and validates the function’s behavior.

Transcript

Please give Ace some feedback