Problem 28 - Applied Calculus

Question

(a) Water is flowing at a constant rate (i.e., constant volume per unit time) into a cylindrical container standing vertically. Sketch a graph showing the depth of water against time. (b) Water is flowing at a constant rate into a cone-shaped container standing on its point. Sketch a graph showing the depth of the water against time.

Step-by-Step Solution

Answer

For the cylinder, depth vs. time is a straight line; for the cone, it's an upward curving line.

Step 1: Understanding a Cylindrical Container

When water flows into a cylindrical container at a constant rate, the depth (or height) of the water increases linearly with time. This is because the volume of water entering per unit time is constant and so is the cross-sectional area of the cylinder's base. As a result, the relationship between depth and time is linear.

Step 2: Sketching the Graph for the Cylinder

For a cylindrical container, plot time on the x-axis and depth on the y-axis. The graph will be a straight line with a positive slope, because each moment in time corresponds to an equal increase in depth due to the constant inflow rate. This straight line indicates that the height of water increases steadily over time.

Step 3: Understanding a Cone-shaped Container

In a cone-shaped container, as water flows in at a constant rate, the depth increases more slowly at first and more quickly as the cone fills up. This is because the volume of a cone varies with the square of the radius as it fills, while the inflow rate is constant. As the depth increases, a larger radius results in a larger area, causing a nonlinear depth increase.

Step 4: Sketching the Graph for the Cone

For the cone-shaped container, plot time on the x-axis and depth on the y-axis. The graph will be a curve that starts off shallow and gradually steepens. Initially, small increases in depth occur slowly due to the narrow base, but as the radius becomes larger, the depth increases more rapidly reflecting the increase in volume with the square of the depth.

Key Concepts

Cylindrical Container

A cylindrical container is essentially like a vertical tube with a constant cross-sectional area across its height. When water is poured into such a container at a constant rate, the volume of water that accumulates over a given period is directly related to the height of the water. This is because the base area's size remains unchanged, leading to a consistent rise in water level.
Understanding this concept can be simplified to a real-world experience like filling a tall cylindrical glass with water. As you pour steadily, the water level rises at an even pace, as long as the pouring speed is constant. This is why, in mathematical terms, if you graph the depth of water against time, you get a straight line. This line directly illustrates how, for each increment of time, there's an equal increase in water depth due to the uniform inflow volume.

Cone-shaped Container

A cone-shaped container presents a fascinating geometric challenge to understand. Unlike the cylinder, a cone widens as it gets taller. Imagine pouring water into an ice cream cone point down. Initially, when water enters, it barely rises because the narrow tip can hold only a small amount of water. This changes as it fills further up, where the cone widens, allowing more water to result in a faster rise in depth.
In practical terms, this means that when you graph this relationship on paper, it starts off as a gentle curve and progressively becomes steeper. This gradual change in the graph's slope is because while the water inflow remains constant, the base area getting larger results in more dramatic increases in water depth as time goes on.

Graph Sketching

Graph sketching is a powerful visual tool we use to understand relationships like those presented by these models. Drawing effective graphs involves understanding the variables at play. For our examples, the x-axis represents time, and the y-axis represents depth. The task is to accurately reflect how each container's water depth changes over time.
When graphing a cylindrical container, the graph is straightforward—a line moving directly upward at a constant rate. For the cone-shaped container, however, the drawing starts with a softer curve that rises more severely as time progresses.
This difference in graphical representation provides a clear illustration of the underlying physical realities of both container shapes. Consistent volume affects each shape differently based on their designs.

Depth and Time Relationship

Depth and time relationship is a critical point in understanding geometric containers' behaviors. This relationship changes dramatically based on the shape of the container. In the cylindrical shape, depth changes uniformly with time due to consistent area, creating a linear depth-time graph.
For cone-shaped containers, as they fill, their cross-sectional area gets wider, causing a depth-time relationship that evolves from slow to fast changes—captured by a curving graph. This reflects more profound mathematical principles, as the formula for a cone's volume, \( V = \frac{1}{3}\pi r^2 h \), shows reliance on changing radius proportions compared to a cylinder's consistent \( V = \pi r^2 h \).
By thoroughly understanding these relationships, students can grasp how different shapes and constant inflows interact, an important concept in physical and applied mathematics.