Problem 61 - Calculus with Concepts in Calculus

Question

Express the height \(h\) of a right circular cylinder as a function of the volume \(V\) and radius \(r\).

Step-by-Step Solution

Answer

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

Step 1: Recall the Volume Formula for Cylinder

The volume of a right circular cylinder is given by the formula: \[ V = \pi r^2 h \]where \( V \) is the volume, \( r \) is the radius, and \( h \) is the height of the cylinder.

Step 2: Solve for Height

We need to express height \( h \) as a function of \( V \) and \( r \). Start by solving the volume formula for \( h \):\[ V = \pi r^2 h \] Divide both sides by \( \pi r^2 \) to isolate \( h \):\[ h = \frac{V}{\pi r^2} \]

Step 3: State the Function

Thus, the height \( h \) of the cylinder can be expressed as a function of the volume \( V \) and the radius \( r \):\[ h(V, r) = \frac{V}{\pi r^2} \]

Key Concepts

Right Circular Cylinder

A right circular cylinder is a common geometric shape whose circular base is perpendicular to its height. This geometry is often seen in everyday objects like cans or tubes. Understanding the structure of a right circular cylinder helps in calculating various properties.

• The base of this cylinder is a circle. The radius \(r\) represents the distance from the center of the base to any point on its perimeter.
• The height \(h\) is the measure from the base to the top, along a straight line parallel to the sides.

The volume of a right circular cylinder can be easily calculated using the radius and height, demonstrating the utility of this shape in calculation and design.

Function of Two Variables

Expressing the height of a right circular cylinder in terms of volume and radius means creating a function with these two as its variables. In mathematics, a function is a relationship where each input (in this case, pairs of \(V\) and \(r\)) corresponds to exactly one output \(h\).

• The volume \(V\) of the cylinder gives an idea of how much space it takes up or can contain.
• The radius \(r\) helps determine both the size of the base and indirectly contributes to the volume.

By rearranging the volume formula, you achieve a function \(h(V, r) = \frac{V}{\pi r^2}\) that illustrates how the height changes with different volumes and radii, providing a way to predict dimensions given certain conditions.

Geometry Formula

Formulas in geometry provide tools to calculate various properties of shapes. For the right circular cylinder, the formula for volume \( V = \pi r^2 h \) plays a critical role. This formula combines basic geometric principles, involving the area of a circle and straight height multiplication, to find the cylinder's volume.

• The term \(\pi r^2\) calculates the area of the circular base. \(\pi\) (approximately 3.14159) is a constant relating circle circumference to its diameter.
• The height \(h\) extends this area into three dimensions, multiplying how much space the base shape occupies upwards.

Understanding this formula allows for flexibility in solving for any missing component—whether it's height, radius, or volume—when you have the other two. Depth in grasping this concept supports more advanced applications of geometry in both academic and real-world contexts.