MM - Other Function Models Lesson

Other Functions Models

Hurricanes in Texas

Texas experiences a wide variety of weather, including hurricanes. Coastal residents often feel the direct effects of hurricanes when they make landfall along the coast. Cities and towns that are directly hit by a hurricane can sometimes take years to rebuild. Galveston is one such city.

Galveston was almost completely destroyed by the storm that hit in 1900, the deadliest hurricane in U.S. history. Rebuilding after the storm took several years, partly because residents raised the elevation of the entire city and built the Galveston Seawall to protect the city. Other towns were not so resilient. In 1886, residents of Indianola completely abandoned the ruins of their town on the shores of Matagorda Bay after it was wiped away by a strong hurricane.

Meteorologists use the Saffir-Simpson scale to describe the strength of a hurricane. This scale is based on a combination of wind speed and barometric pressure. The faster the wind speed and the lower the barometric pressure, the higher the rating of the hurricane on the Saffir-Simpson scale.

Many hurricanes have struck the Texas coast, but there have been no recorded Category 5 hurricanes, which are the strongest, most destructive storms. Although many Caribbean and Central American nations have been pounded by Category 5 hurricanes, the United States has been hit by only three: the 1935 Labor Day Hurricane, which struck the Florida keys; Hurricane Camille, which struck Pass Christian, Mississippi, in 1969; and Hurricane Andrew, which struck near Homestead, Florida, in 1992.

The following table shows the year, wind speed, and Saffir-Simpson category for some hurricanes that have made landfall on the Texas coast. This table also includes the Category 5 storms that have hit the United States.

Let's write a dependency statement that describes the relationship between the two variables, wind speed and Saffir-Simpson category.

Answer: The Saffir-Simpson category depend on the wind speed of the hurricane. 

Now, let's create a scatterplot of the Saffir-Simpson category versus wind speed for the hurricanes listed in the table.

Next, mark the wind speed endpoints for each Saffir-Simpson category on the scatterplot.

Now, let's connect those endpoints with a line segment. For example, along the line for Category 1, mark the wind speeds 74 and 95 [that is, the points (74, 1) and (95, 1)] - connect with a line segment.

Is it possible for a hurricane to be rated between Category 1 and Category 2? Why or why not?

Answer: No. A Category 1 hurricane has winds between 74 and 95 miles per hour, and a Category 2 hurricane has winds between 96 and 110 miles per hour. There is not a category for a hurricane with a wind speed of between 95 and 96 miles per hour.

Hurricane wind speeds are difficult to measure precisely. Thus, most hurricane wind speeds are estimated to the nearest 5 miles per hour. Suppose a new technology were invented that allowed meteorologists to measure hurricane wind speeds very precisely.

If a hurricane had a wind speed of 95.1 miles per hour, what category would it be rated? How do you know?

Answer: According to the Saffir-Simpson scale, there is no way to categorize a hurricane with a wind speed of between 95 and 96 miles per hour.

Revise the Saffir-Simpson scale so that you can rate hurricanes with wind speeds that lie between the existing categories.

When graphing inequalities, how do you represent an endpoint that does not include or equal to?

Answer:

Use an open circle to represent up to but not including.

Use a closed or open endpoint to revise your scatterplot for the new hurricane rating scale.

The type of function you graphed is called a step function.

The graph is a series of segments and look like small steps.

Concentrations of Medicine

Have you ever taken a medication that your doctor warned you would not take effect for a few days? In this activity, you will investigate why that is the case.

Consider the allergy medicine "Sneeze-B-Gone." The regular adult dose is 20 milligrams. As with all medicines, the body gradually filters Sneeze-B-Gone out of the bloodstream. The rate at which the medicine is filtered out is called the flush rate. For Sneeze-B-Gone, the flush rate is 30%. In other words, 24 hours after the pill is taken, 30% of Sneeze-B-Gone has flushed out of the body.

Let's review.  Try to answer the questions below independently. Answers are provided for you to check your results.

1. If 30% of Sneeze-B-Gone has flushed out of the body after 24 hours, what percent of Sneeze-B-Gone remains?

Answer:

100% - 30%=70%

Use your calculator's recursion feature to create table, assuming that an adult is taking one 20-milligram dose per day.

2. At what value does the amount of Sneeze-B-Gone in the bloodstream level off? How many days does it take for that to happen?

Answer:

66.6 milligrams; about 18 days

3. What type of function could model the amount of Sneeze-B-Gone in the bloodstream as a function of time?

Answer:

Since each day of medication contains a rate of decay, the behavior of an exponential function is expected. The patient, however, also takes a new pill daily, which increases the amount of Sneeze-B-Gone in the bloodstream quickly. Therefore, discontinuous behavior is also expected, which leads to a type of step function.

Recall that the general form for exponential decay functions is y = a(b) x , where a represents the starting amount of the substance and b represents either the growth factor or factor of decay. For growth, b=1+r and for decay, b=1-r, where r is the decimal form of the percentage of rate of growth or decay. For a 20-milligram dose and a 30% flush rate, what exponential function could describe the amount of Sneeze-B-Gone in the bloodstream (y) as a function of time (x)? (Do not forget that b represents the percent of Sneeze-B-Gone that remains in the bloodstream.)

Since the patient did not begin taking the medicine until Day 1, adjust your function rule by subtracting 1 from the exponent. Graph the function on your graphing calculator. Sketch your graph and describe your viewing window.

 

The viewing window: 0 < x < 7 on the x-axis and 0 < y < 70 on the y-axis.

If time (x) is given in terms of the number of days, what happens to the amount of Sneeze-B-Gone in the patient's bloodstream at the start of Day 2 when the patient takes a second pill? How does this affect the graph?

The amount increases by the new dose, or 20 milligrams. This causes a vertical shift or jump in the graph at Day 2, suggesting the need for a step-function-style graph.

Let's restrict the domain to show the jump.

There are several ways to restrict the graph using a T1 graphing calculator. One way to restrict the graph is 

For Day 2, enter the function  into your calculator. What do the constants 34, 0.7, and 2 represent? Sketch the new graph.

34: the amount of Sneeze-B-Gone in the bloodstream at the beginning of Day 2

0.7: the percent of Sneeze-B-Gone remaining in the bloodstream after 24 hours (rate of decay)

2: a horizontal shift in the domain for Day 2

Based on the functions for Day 1 and Day 2,write a function from the data in your table for Day 3 and a function for Day 4.

Day 3:

Day 4:

Graph both of these new functions. What patterns do you notice? What do you expect the graph for Day 5 to look like?

 

4. Assume the patient takes 20 milligrams of Sneeze-B-Gone every day. If you extend the graph to Day 20 or beyond, what would the minimum amount of Sneeze-B-Gone in the bloodstream be? The maximum amount?

Answer:

The amount of Sneeze-B-Gone in the bloodstream stays at 66.6 milligrams. The graph becomes a horizontal line.

 

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