Problem 101 - A Graphical Approach to College Algebra

Question

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.

Step-by-Step Solution

Answer

Volume gained is approximately 1174.49 cubic inches.

Step 1: Understanding the Formula

The formula for the volume of a sphere is given by \( V = \frac{4}{3} \pi r^3 \), where \( r \) is the radius. We need to find the volume of a sphere with radius \( r + 3 \) and compare it to the original volume with radius \( r \).

Step 2: Finding Function D(r)

The volume of a sphere with radius \( r + 3 \) is \( V(r+3) = \frac{4}{3} \pi (r+3)^3 \). The original volume is \( V(r) = \frac{4}{3} \pi r^3 \). Therefore, the volume gained, \( D(r) \), is given by \( D(r) = V(r+3) - V(r) = \frac{4}{3} \pi ((r+3)^3 - r^3) \).

Step 3: Simplifying D(r)

To simplify, calculate \( (r+3)^3 - r^3 \). Expanding \( (r+3)^3 \) gives \( r^3 + 9r^2 + 27r + 27 \). Thus, \( D(r) = \frac{4}{3} \pi (9r^2 + 27r + 27) \).

Step 4: Setting up the Graph

To graph \( y = D(r) \) in the window \([0,10]\) by \([0,1500]\), use a graphing calculator or software. Set the x-axis to represent the radius from 0 to 10 and the y-axis to represent the volume gained from 0 to 1500.

Step 5: Using the Calculator to Estimate

Using the calculator, plot the function and find the y-value when \( x = 4 \). This gives the volume gained when the radius increases from 4 inches to 7 inches (since 4 + 3 = 7).

Step 6: Verification of Result

To verify, calculate \( D(4) \) analytically. First find \( D(r) = \frac{4}{3} \pi (9 \times 4^2 + 27 \times 4 + 27) \), evaluate this expression to find the exact volume gained. Simplifying gives \( D(4) = \frac{4}{3} \pi (144 + 108 + 27) = \frac{4}{3} \pi \times 279 \).

Step 7: Final Calculation and Verification

Calculate \( \frac{4}{3} \pi \times 279 \) to find the volume gained. This numeric result should match the estimate from your graph.

Key Concepts

Volume Formula

The volume of a sphere relies on a simple but powerful formula: \( V = \frac{4}{3} \pi r^3 \). This formula lets us determine the space inside a spherical object based on its radius, \( r \). To find the volume, you plug in the sphere's radius and follow through the equation; that is, cube the radius, multiply by pi, and scale by the fraction \( \frac{4}{3} \).

When we're dealing with problems that involve changing the size of a sphere, such as increasing the radius, this formula is the backbone of our calculations. By knowing the original and new radius, one can calculate and compare the volumes easily using this formula.

Function Notation

Function notation makes complex calculations easier and more intuitive. In the exercise, the function \( D(r) \) represents the volume gained by increasing a sphere's radius by 3 inches.

Instead of calculating volumes twice and subtracting them, you can express this entire process as a function:

• Calculate the volume of a sphere with radius \( r+3 \): \( V(r+3) = \frac{4}{3} \pi (r+3)^3 \).
• Calculate the original volume \( V(r) = \frac{4}{3} \pi r^3 \).
• The difference gives \( D(r) = \frac{4}{3} \pi ((r+3)^3 - r^3) \).

By introducing \( D(r) \), it becomes clear how much additional volume results from the increase in radius, making the calculation both simplified and precise.

Graphing Calculator

Graphing calculators or tools like software applications can visually represent functions like \( D(r) \). When you plug \( D(r) = \frac{4}{3} \pi (9r^2 + 27r + 27) \) into a calculator and set the graph to cover the range specified (from 0 to 10 on the x-axis and 0 to 1500 on the y-axis), it displays how the sphere's volume gain changes as the radius varies.

Using this tool allows us to graphically depict relationships and observe patterns that might not be obvious from raw numbers. For instance, if you're curious about the increased volume when the radius changes from 4 inches to 7 inches, the calculator plots the exact y-value at \( x = 4 \), which shows the volume gain graphically.

Analytical Verification

Analytical verification confirms the accuracy of results obtained through estimation or graphing. In this exercise, once the graph gives an estimate for the volume gain as the radius increases from 4 to 7 inches, you can verify this result with direct calculation.

Compute \( D(4) \) step by step:

• Expand \( D(r) = \frac{4}{3} \pi (9 \times 4^2 + 27 \times 4 + 27) \).
• Calculate inside the parentheses: \( 9 \times 16 + 108 + 27 = 279 \).
• Conclude with \( D(4) = \frac{4}{3} \pi \times 279 \), providing a precise value.

With solid numerical proof, analytical verification clarifies and confirms what you observe graphically, ensuring the reliability of your results.