Lesson Plan #6



Lesson Plan #021

Class: PreCalculus Date: Tuesday October 24th, 2014

Topic: Mathematical Modeling

Aim: How do we use mathematical models to approximate sets of data points?

Objectives:

1) Students will be able to use mathematical models to approximate sets of data points.

2) Students will be able to solve direct, joint, and inverse variation problems.

HW# 021: Page 124 #’s 8, 12, 16, 20, 26, 30, 36, 42,

Page

Do Now:

The table at right lists the total tuition, room and board rates charged for

full-time undergraduate students in degree-granting institutions. Let’s plot these points

on a graphing calculator. We’ll use 1 for year 2001.

On your graphing calculator press the [STAT] button. On the Edit Menu, press 1. Edit. Move the cursor to the top of L1. Press [CLEAR] and press . Do the same for L2. Go back to L1, in the first line for data enter 1 then down and enter 2, so on and so forth until you get to 11. Starting at the top of L2, enter the cost of tuition; for $13,393 enter 13.393. Go to [Y=] and turn off any equations there from graphing. Then press 2nd [Y=] which takes you to STAT PLOT. Make sure it is ON, points, XList: L1, YList: L2, Mark is the little box.

Then Zoom 9 to see the plots.

What kind of graph would be a good for the plotted points? (linear, quadratic, exponential, etc.)?

Procedure:

Write the AIM and DO NOW

Get students working!

Take attendance

Give back work

Go over HW

Collect HW

What is the equation?

Example:

In Pennsylvania, the state income tax is directly proportional to gross income. You are working in Pennsylvania and your state income tax deduction is $46.05 for a gross monthly income of $1500. Find a mathematical model that gives the Pennsylvania state income tax in terms of gross income.

If a mathematical model follows a model such [pic], this model is referred to as direct variation model.

Exercise #1: Variable M varies directly with p. If M = 75 when p = 10, find M when p = 16.

Exercise #2: Variable Y varies jointly with P and Q.   If Y = 144 when P = 12 and Q = 8, find Y when P = 15 and 

Q = 25.

Exercise #3: Variable T varies directly with the square of m. If T is 8 when m = 2, find T when m = 4.

Exercise #4:

The distance a ball rolls down an inclined plane is directly proportional to the square of the time it rolls. During the first second the ball rolls 8 feet. Find a mathematical model that relates the distance traveled to the time . Then find out how far the ball will roll during the first 3 seconds.

Assignment #2: Looking at the table at right:

1) What do you notice about the product of x and y?

2) As you go down the columns, what do you notice about the values of x and y?

Inverse Variation:

For two quantities with inverse variation, as one quantity increases, the other quantity decreases.

Also the product between the two variables is always constant.

Exercise #5: R varies inversely with variable T.  If R is 168 when T = 24, find R when T = 30.

Exercise #6: A video store rents DVDs, and the weekly total number varies directly with the total inventory, and varies inversely with the cost of each rental tape.  The store in Acme, NY, last week rented a total of 3000 DVDs when its inventory was 9600 and the cost per rental was $2.40.  If its inventory does not change, what would be the effect on the weekly total number of increasing the cost per rental to $3.00?  

Exercise #7:

Exercise #8:

Hooke’s Law states that the distance a spring is stretched (or compressed) varies directly as the force on the spring. A force of 265 newtons stretches a spring 0.15 meter.

A) How far will the force of 90 netwons stretch the spring?

B) What force is required to stretch the spring 0.1 meter?

On Your Own:

1) A force of 220 netwons stretches a spring 0.12 meter. What force is required to stretch the spring 0.16 meter?

2) Find a mathematical model for the verbal statement.

A) Newton’s Law of Cooling: The rate of change R of the temperature of an object is proportional to the difference between the temperature T of the object and the temperature Te of the surrounding environment.

B) Boyle’s Law: For a constant temperature, the pressure P of a gas is inversely proportional to the volume V of the gas.

C) Newtown’s Law of Universal Gravitation: The gravitational attraction F between two objects of masses m1 and m2 is proportional to the product of the masses and inversely proportional to the square of the distance r between the objects.

3) Find a mathematical model representing the statement. In each case, determine the constant of proportionality.

A) A varies directly as r2. ([pic], [pic])

B) Y varies inversely as x. ([pic]when [pic])

C) F is jointly proportional to r and the third power of s. ([pic]when [pic], [pic][pic])

Sample Test Questions:

1)

2) 3)

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Year Cost

Assignment #1:

Let’s try to find a linear equation that best fits the data.

Go to STAT, CALC, 4:LinearReg(ax+b) L1, L2, Y1 then press . So your equation is now stored in Y1. Use 2nd Calc, then Value to calculate the estimated cost in 2012, 2013 (increase the x maximum to accommodate 2013)

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