Question

The volume \(V\) of a right circular cylinder varies directly with the square of its radius \(r\) and its height \(h .\) The constant of proportionality is \(\pi .\) See the figure. Write an equation for \(V\)

Short answer

The equation for the volume is \( V = \pi r^2 h \).

Step-by-step solution

Understand the relation

The volume of a right circular cylinder varies directly with the square of its radius and its height. This can be expressed as: $$ V \text{ is proportional to } r^2 \text{ and } h $$.

Introduce the constant of proportionality

When a quantity varies directly with another, we introduce a constant of proportionality. Here, the constant of proportionality is given as \( \pi \). So the relationship can be written as: $$ V = \text{constant} \times r^2 \times h $$. Substitute the constant \( \pi \) into the equation: $$ V = \pi \times r^2 \times h $$.

Write the final equation

Now, we have the final equation for the volume of the right circular cylinder: \( V = \pi r^2 h \). This equation represents the volume of a right circular cylinder in terms of its radius and height.

Key concepts

direct variation

In mathematics, direct variation describes a relationship between two variables where one is a constant multiple of the other. In our case, the volume of the cylinder, denoted as \(V\), directly varies with both the square of the radius \(r\) and the height \(h\) of the cylinder. This means if the radius or height increases, the volume increases too, provided all other factors remain constant.

We can write this relationship as:

\[V \propto r^2 \cdot h\]

Here, \(\propto\) means 'is proportional to'. Direct variation is fundamental in connecting physical dimensions to measurable quantities like volume.

radius

The radius \(r\) of a right circular cylinder is the distance from the center of the base to the edge of the base. It's half the diameter of the base. In the formula for volume, the radius is squared, which means it's multiplied by itself. Thus, the volume is highly sensitive to changes in the radius.

If the radius doubles, the square of the radius increases four times, significantly impacting the volume. Hence:

\[r \text{ increases } \Rightarrow r^2 \text{ increases greatly }\Rightarrow V \text{ increases}\]

Understanding the role of the radius is crucial in applications where precise volume calculations are needed.

height

The height \(h\) of a right circular cylinder is the distance between its two bases. This measurement is perpendicular to the bases and often denoted simply as the vertical distance.

The height directly affects the volume linearly. This means if you double the height while keeping the radius constant, the volume also doubles. The relationship can be simplified as:

\[h \text{ increases } \Rightarrow V \text{ increases}\]

Knowing the height is essential because it directly scales the volume and determines the overall capacity of the cylinder.

pi

The constant of proportionality \(\pi\) (pi) is approximately 3.14159 and is a fundamental mathematical constant. It's mostly known for its role in the circumference and area of a circle.

In the volume formula \(V = \pi r^2 h\), \(\pi\) helps relate the circular base (through \(r^2\)) and the height of the cylinder to its volume. This constant ensures that the units of the volume are consistent with the units of \(r\) and \(h\), giving a coherent and unified formula.

Knowing \(\pi\) and its properties helps in accurately calculating the volume of not only cylinders but also other geometrical shapes involving circles.