In this section, real world examples of square root inequalities will be solved. A graphing calculator is needed for this section.

Example: Tsunamis are usually created when there is an earthquake below the ocean floor. Weather forecasters want to be able to predict how fast the tsunami waves are moving. The speed a tsunami is moving depends on the depth of the ocean at that place. There is a square root function that represents this relationship: s = 3.1√d where s is the speed of the wave in meters per second and d is the depth of the ocean in meters. How deep would the ocean have to be to produce a tsunami that was moving faster than 100 meters per second?

Solution

The formula is Interactive button. Assistance may be required. __________ s = 3.1√d and you want to find where s Interactive button. Assistance may be required. __________ > 100

Write an inequality that represents this situation Interactive button. Assistance may be required. __________ 3.1√d > 100

To solve using a graph, you should graph the two sides of the inequality as separate functions.

Your two functions are Interactive button. Assistance may be required. __________ s = 3.1√d and Interactive button. Assistance may be required. __________ s = 100

Since the depth, d, is the independent variable and represents the depth of the ocean floor, what do you know about the values that d could be? Interactive button. Assistance may be required. __________ Depth is positive and since it refers to depth of the ocean it is probably a large number.

Since the speed, s, is the dependent variable and represents how fast a wave is moving, what do you know about the values that s could be? Interactive button. Assistance may be required. __________ Speed is positive and since it refers to the speed of a tsunami wave it is probably a large number.

graphing calculator screen showing the graphs of y = 3.1 times square root of x and y = 100.

When setting the viewing window on your calculator, you need to consider the domain and range of the function.

The functions intersect at the point Interactive button. Assistance may be required. __________ (1041, 100) (rounded to the nearest whole number).

The x-coordinate of this point is Interactive button. Assistance may be required. __________ 1041 which represents the depth in meters of the ocean.

We want to know when the speed is greater than 100, so we want to find the part of the square root graph that lies above the y = 100 line.

According to the graph, the ocean would have to be at least 1041 meters deep to produce a tsunami wave that was moving faster than 100 m/s.

graphing calculator screen showing the graphs of y = 3.1 times square root of x and y = 100 with intersection at x = 1040.5827 and y = 100 with the graph of y = 3.1 times square root of x greater than y = 100 indicated.

Use your calculator to create a table of values for the function.

graphing calculator screen showing a table of values for X, Y1, and Y2 from x = 700 to x = 1300

X represents the depth
Y1 represents the speed

graphing calculator screen showing a table of values for X, Y1, and Y2 from x = 700 to x = 1300 with x = 100 and x = 1100 indicated.

Notice that the Y1 values jump from 98.031 to 102.82 between X = 1000 and 1100.

Now you can change the increment that your table counts back to 1, starting anywhere between x = 1000 and 1100.

 

According to the table, the ocean would have to be at least 1041 meters deep to produce a tsunami wave that was moving faster than 100 m/s.

Note: x = 1040 is close to but not greater than 100.