Concept explainers
Torricelli's Law Water in a tank will flow out of a small hole in the bottom faster when the tank is nearly full than when it is nearly empty. According to Torricelli’s Law, the height h(t) of water remaining at time t is a quadratic function of t.
A certain tank is filled with water and allowed to drain. The height of the water is measured at different times as shown in the table.
(a) Find the quadratic polynomial that best fits the data.
(b) Draw a graph of the polynomial from part (a) together with a
(c) Use your graph from part (b) to estimate how long it takes for the tank to drain completely.
Time (min) | Height (ft) |
0 | 5.0 |
4 | 3.1 |
8 | 1.9 |
12 | 0.8 |
16 | 0.2 |
(a)
To find: The quadratic polynomial that best fits the data.
Answer to Problem 6P
The quadratic polynomial that best fits the data is
Explanation of Solution
It is appropriate to use quadratic polynomial to model the given data, if there is single peak in the given data.
It is observed that the given data appears to have a peak. Therefore, it is appropriate to use quadratic polynomial (degree 2) as a model for the given data
By the use of graphing calculator, the best fit quadratic regression
Thus, the quadratic polynomial that best fits the data is
(b)
To draw: The graph of the quadratic polynomial from part (a) with the scatter plot of the data.
Explanation of Solution
Graph:
Consider the time in minutes as the x coordinates and the height in feet as the y coordinates.
From part (a), the quadratic polynomial obtained is
The graph of the quadratic polynomial with the scatter plot of the given data is shown below in Figure 1.
From Figure 1, the graph is an upward parabola.
(c)
To estimate: The time for the tank to drain completely.
Answer to Problem 6P
The time for the tank to drain completely is
Explanation of Solution
The tank drains completely when the height of water remaining in the tank is zero.
From Figure 1, it is observed that the value of y is 0 occurs when x is 18.68.
Therefore, the time for the tank to drain completely is
Chapter 3 Solutions
Precalculus: Mathematics for Calculus - 6th Edition
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- Calculus: Early TranscendentalsCalculusISBN:9781319050740Author:Jon Rogawski, Colin Adams, Robert FranzosaPublisher:W. H. FreemanCalculus: Early Transcendental FunctionsCalculusISBN:9781337552516Author:Ron Larson, Bruce H. EdwardsPublisher:Cengage Learning