Problem 75 - A Graphical Approach to Precalculus with Limits

Question

Solve each formula for the specified variable. \(V=\frac{1}{3} \pi r^{2} h\) for \(h \quad\) (Volume of a cone)

Step-by-Step Solution

Answer

The formula for \( h \) is \( h = \frac{3V}{\pi r^2} \).

Step 1: Understand the Formula

The given formula is for the volume of a cone: \( V = \frac{1}{3} \pi r^2 h \). This formula calculates the volume based on the radius \( r \), height \( h \), and pi (approximated as 3.14159). Our task is to solve this equation for \( h \).

Step 2: Isolate the Variable

To solve for \( h \), we need to isolate \( h \) on one side of the equation. To do this, start by multiplying both sides by 3 to eliminate the fraction, giving \( 3V = \pi r^2 h \).

Step 3: Divide to Solve for \( h \)

Now, divide both sides of the equation by \( \pi r^2 \) to solve for \( h \). The equation becomes \( h = \frac{3V}{\pi r^2} \).

Step 4: Final Expression

Now that \( h \) is isolated, the expression \( h = \frac{3V}{\pi r^2} \) is the solution to the problem. This equation tells us how to calculate the cone height given its volume and radius.

Key Concepts

Volume of a Cone

The volume of a cone is a fundamental concept in geometry, relating the space it occupies to its shape dimensions. To calculate the volume of a cone, you use the formula:\[ V = \frac{1}{3} \pi r^2 h \]Where:

• \(V\) is the volume, which tells you how much space is inside the cone.
• \(\pi\) is a constant, approximately 3.14159, representing the ratio of the circumference of a circle to its diameter.
• \(r\) is the radius, the distance from the center to the edge of the base.
• \(h\) is the height, the straight line distance from the base to the apex.

This formula shows that the volume of a cone is one-third of what it would be if it were a cylinder with the same base and height. Understanding this formula helps in solving many practical problems such as determining the capacity of conical containers.

Algebraic Manipulation

Algebraic manipulation is a crucial skill in solving equations and expressions. It involves strategically altering equations to isolate desired variables or simplify expressions. In the context of our problem, we are required to solve for the height \(h\) from the volume equation.Given:\[ V = \frac{1}{3} \pi r^2 h \]To isolate \(h\), follow these steps:

• First, multiply both sides of the equation by 3 to counteract the fraction, resulting in \(3V = \pi r^2 h\).
• Next, divide both sides by \(\pi r^2\) to solve for \(h\), resulting in \(h = \frac{3V}{\pi r^2}\).

This manipulation isolates \(h\) and provides the necessary equation to calculate the height based on given volume and radius. Practicing these steps will strengthen your algebra skills, enabling you to tackle more complex mathematical challenges.

Geometry Concepts

The problem of finding the height of a cone based on its volume and radius dives into important geometry concepts. Geometry is fundamentally about the properties and relation of points, lines, surfaces, and solids.Key elements of a cone:

• **Base:** A cone has a circular base that provides the area \(\pi r^2\).
• **Height:** The height \(h\) is the vertical distance from the base to the tip of the cone.
• **Slant Height:** Often in geometry, we discuss the slant height, which is different from height; it is the distance along the side of the cone.

Understanding these elements and how they interrelate is crucial. This problem also illustrates how volume connects to the geometry concepts and helps to solve practical tasks. By learning these concepts, you can solve more varied and complex geometric problems.