Problem 37 - Precalculus Mathematics for Calculus

Question

Solve the equation for the indicated variable. $$V=\frac{1}{3} \pi r^{2} h ; \quad \text { for } r$$

Step-by-Step Solution

Answer

The solution is \( r = \sqrt{\frac{3V}{\pi h}} \).

Step 1: Understanding the Problem

We need to solve the equation \( V = \frac{1}{3} \pi r^2 h \) for the variable \( r \). This means we want to isolate \( r \) on one side of the equation.

Step 2: Eliminate the Fraction

To remove the fraction, multiply both sides of the equation by 3 to get rid of the denominator. This gives us \( 3V = \pi r^2 h \).

Step 3: Isolate the \( r^2 \) Term

Divide both sides by \( \pi h \) to isolate \( r^2 \) on one side: \( \frac{3V}{\pi h} = r^2 \).

Step 4: Solve for \( r \)

Take the square root of both sides to solve for \( r \): \( r = \sqrt{\frac{3V}{\pi h}} \).

Key Concepts

Volume Formula

The volume formula is a mathematical equation used to find the space occupied within a 3-dimensional object. In our original problem, the formula is given for the volume of a cone: \( V = \frac{1}{3} \pi r^2 h \). This formula combines several elements:

• \( V \) (Volume): The amount of space the cone occupies.
• \( r \) (Radius): The distance from the center to the edge of the circular base of the cone.
• \( h \) (Height): The perpendicular distance from the base to the tip of the cone.

The formula uses \( \pi \), a constant approximately equal to 3.14159, and represents the ratio of the circumference of a circle to its diameter.
In this case, the formula incorporates multiplying the area of the circular base, \( \pi r^2 \), by the height \( h \) and then dividing by 3 since a cone is essentially one-third of a cylinder with the same base and height.

Shape Geometry

Shape geometry provides the background knowledge needed to understand how the volume formula shapes the equation we are solving. Here we focus on the shape of a cone, which is a 3-dimensional figure with a circular base that tapers smoothly to a point, known as the apex. Geometry helps us understand the spatial relationships between different parts of the cone.

• Cones have a circular base, which means their base geometry is pivotal in calculating the volume.
• The height, an essential component, affects the total volume. It measures the vertical space the cone occupies.
• The symmetry and proportions of a cone guide how and why the volume formula leverages squared radius values.

Understanding this geometry helps when isolating variables and manipulating the volume formula, as you'll see when we solve for radius, \( r \). A solid comprehension of these geometric principles aids in visualizing the problem, which is pivotal for problem-solving.

Isolating Variables

Isolating variables is a key skill in solving algebraic equations, such as our task to find the radius \( r \) in the volume formula. This process involves re-arranging the equation so that one variable stands alone on one side of the equality:

• Begin by eliminating unwanted components around the variable you need to isolate. In our case, we first remove the fraction by multiplying through by 3.
• Next, bring the desired variable \( r \) closer to isolation by dividing away coefficients 'attached' to it, here, multiplying by \( \pi \) and \( h \).
• To completely isolate \( r \), take the square root of both sides to solve for \( r \) given \( r^2 \).

When isolating variables, it's crucial to perform inverse operations to simplify and solve equations correctly. This meticulous following of mathematical rules ensures that you accurately find the desired variable, \( r \), in this instance from the cone's volume formula.