1
Example 1
Consider the tank in a shape of an inverted circular cone where the tanks radius is 3 feet and the depth is 5 feet. The tank is being drained through a hole in the vertex at a rate of 4 cubic feet per second.
a. Sketch a labeled picture of the tank, including a snapshot of there being water remaining in the tank prior to it being empty.
b. What are some quantities that are changing in this scenario? What are some quantities that are not changing?
c. The volume of an inverted circular cone is [latex]V = \frac{1}{3}h \pi r^2[/latex]. When the tank is completely full how much water is present?
d. How long will it take for the tank to drain completely?
e. What is the height of the water when the tank is full?
f. Sketch possible graphs of how [latex]V[/latex] and [latex]t[/latex] change in tandem and how [latex]h[/latex] and [latex]t[/latex] change in tandem. For each graph label any ordered pairs you know for certain and write at least one sentence to describe the behavior of the graph.
Show Solution
a.
b. Changing: the height of the water in the tank, the volume of the water in the tank, and time.
Not changing: the rate at which the water leaves the tank, the shape of the tank.
c. When the tank is full [latex]h = 5[/latex] and [latex]r = 3[/latex], so
\begin{align*}
V & = \frac{1}{3} \cdot 5 \cdot \pi \cdot 3^2 \\
V & = \frac{1}{3} \cdot 45\pi = \frac{45\pi}{3} \\
V & \approx 47.12 ft^3 \\.
\end{align*}
d. Since the tank is draining 4 cubic feet per minute, then the tank will take [latex]\frac{47.12}{4} \approx 11.78[/latex] minutes to drain.
e. The height of the water when the tank is full is 5 feet.
f.
Since the volume is decreasing at a constant rate over time the graph should have a linear relationship. For the second graph, since the tank is skinnier near the bottom of the tank then the height of the water will move slower at first and over time as it reaches the end the height of the water will decrease faster.
Example 2
The graph below shows the average monthly high temperature in Durango over a twelve month period. Let January be month 1.
a. What is the warmest month in Durango?
b. Over what interval of months is the temperature increasing? What interval of months is the temperature decreasing?
Show Solution
a. The warmest month in Durango is July.
b. The temperature is increasing from January to July and then decreasing from July to December.