Problem 49 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

During road construction gravel is being poured onto the ground from the top of a tall truck. The gravel falls into a conical pile whose height is always equal to half of its radius. Express the amount of gravel in the pile as a function of its (a) height. (b) radius.

Step-by-Step Solution

Answer

The amount of gravel in the pile as a function of its height is \[V = \frac{2}{3} \pi h^3\]. The amount of gravel as a function of its radius is \[V = \frac{1}{6} \pi r^3\].

Step 1: Substitute the relationship between radius and height into the volume formula

Substitute \(r = 2h\) into the formula:\[V = \frac{1}{3} \pi r^2 h\]to get\[V = \frac{1}{3} \pi (2h)^2 h = \frac{2}{3} \pi h^3\]

Step 2: Express the volume in terms of radius

Substitute \(h = \frac{r}{2}\) into the formula:\[V = \frac{1}{3} \pi r^2 h\]to get \[V = \frac{1}{3} \pi r^2 (\frac{r}{2}) = \frac{1}{6} \pi r^3\]

Key Concepts

Volume of a Cone

Understanding the volume of a cone is crucial when dealing with problems related to three-dimensional shapes. A cone is a geometric solid that has a circular base and a single vertex, forming a shape similar to an ice cream cone. The formula to calculate the volume of a cone is given by:
\[ V = \frac{1}{3} \pi r^2 h \]
where \( V \) represents the volume, \( r \) is the radius of the base, and \( h \) is the height of the cone. This formula comes from calculus and the concept of integration, which allows for the calculation of volumes of various shapes. The one-third factor in the cubic volume formula of a cone is because it is precisely one-third of the volume of a cylinder with the same base and height.
To remember this formula, you might visualize filling up the cone with liquid three times to match the volume of a cylinder that encases it. This can be particularly helpful when envisioning problems like the conical gravel pile in our exercise.

Conical Pile in Calculus

Calculus often deals with problems involving rates of change and how shapes can be formed or altered over time, such as a conical pile of gravel during construction. Considering the pile grows in such a way that its height is always half its radius; this introduces a relationship between the shape's dimensions.
This relationship is essential to solving many real-world problems using calculus, as it can simplify the calculations required. For example, in the provided exercise, we obtain a new formula for volume that is a function of either height or radius.
Using calculus, we translated the physical conditions into mathematical equations, and by substituting our given relationships into the volume formula, we are able to express the amount of gravel in terms of the pile's height (\( h \)) or radius (\( r \)), which are often more easily measured or observed in real construction scenarios.

Relating Volume to Radius and Height

In many practical situations, like the example used in the textbook exercise, there's often a direct relationship between the dimensions of an object. For the cone, specifically, knowing how the volume changes with either the radius or the height is important.
In the conical pile of gravel, the height is set to be always half the radius. Expressing one dimension in terms of the other (the height in terms of the radius or vice versa) is crucial in creating a function to describe the volume.
In our exercise, by substituting the height in terms of the radius into the volume formula, we obtained \( V = \frac{1}{6} \pi r^3 \), and by substituting the radius in terms of the height, we got \( V = \frac{2}{3} \pi h^3 \). These formulas enable the calculation of the gravel pile's volume as it grows over time, based on easily measurable quantities, which can be extremely valuable in fields like construction planning and resource allocation.