Subjects
Grades
Solving Problems Involving the Square Root Function
Concept sheet | Mathematics
Secondary
4-5
To solve a problem involving a square root function, it is important to understand all the inner workings of the function.
Example Where |h=0| and |k=0|
An experiment on falling objects is conducted. After several attempts, it becomes clear that the relationship between the time it takes for an object to reach the ground and the height from which it is released is a square root function. In all of the tests, air resistance was negligible.
a) If the object is thrown from the top of the Empire State Building in New York, which has a height of |381\ \text{m}|, and air resistance is negligible, how long will the fall be?
b) If the object is thrown from the top of the Cliffs of Moher, a beautiful Irish cliff, and the object takes |6.61\ \text{sec}| to hit the water, what was the height of the cliff?
Here is a graph which illustrates the first results of the experiment.
See solution
The next example addresses the concept of free fall and linear motion with constant acceleration, but this time uses a formula similar to those used in physics.
Example Where |h| and |k| are Non-Zero
Scientists drop a free falling object to make precise measurements. The air resistance is considered negligible. The equation that gives the time from the moment the object is dropped |(t)| as a function of the height of the object |(h)| from the ground is the following. ||t=\sqrt{\frac{-2}{g}(h-h_0)}|| where |g| is the gravitational acceleration, approximately |9.81 \text{ m}/\text{s}^2| on Earth, and |h_0| is the initial height in metres, i.e. the height (from the ground) at which the object is released at |t=0| s.
a) How high was the object released if it hit the ground after |2.473| seconds?
b) If the object is at a height of |10| m after |2.995| seconds, after how long was it at a height of |20| m? How high is the object exactly |1| second before it hits the ground?
See solution
In the previous example, there were only equations and not inequalities. A final question with inequalities could have been: "During what time interval is the object at a height of less than |15| metres?" Or, "What is the height of the object when the elapsed time is |2.5| seconds or more?"
To learn how to answer these kinds of questions, see the following concept sheet for more details: Solving a Square Root Equation or Inequality.