# Mathematics

**Lesson 3.6**

**Introduction**

**Course Objectives**

This lesson will address the following course outcomes:

· 20. Translate problems from a variety of contexts into mathematical representation and vice versa (linear, exponential, simple quadratics).

· 22. Identify when a linear model is reasonable for a given situation and, when appropriate, formulate a linear model. In the context of the situation interpret the slope and intercepts and determine the reasonable domain and range.

· 25. Use functional models to make predictions and solve problems.

**Specific Objectives**

Students will understand that

· linear models can be created based on collected or given data.

Students will be able to

· write the equation for a linear model given a set of data.

**Spring Scale**

**Problem Situation 1: Spring Scale**

Watch this demonstration of hanging weights on a spring.

https://www.youtube.com/watch?time_continue=4&v=Dfrl-EcU0kQ

The video gave this data:

mass (grams) |
50 | 100 | 150 |

height (cm) |
25.8 | 21 | 16.2 |

#1 Points possible: 5. Total attempts: 5

Using the dot tool, plot all three points from the video on the graph below. Then, switch to the line tool and draw a line that passes through the points. Mass is on the horizontal axis and height is on the vertical axis.

Clear All Draw:

#2 Points possible: 5. Total attempts: 5

Write an equation for the line giving the height, *h*, when the mass of the weights added to the spring is *w*.

#3 Points possible: 15. Total attempts: 5

The equation tells us that when no weight is on the hanger, the height will be , and that as weight is added, the height by

#4 Points possible: 5. Total attempts: 5

The mystery block of wood pulled down the hanger to 24.5 cm. Use this with your linear model to estimate the mass of the block, rounded to 1 decimal place.

grams

**Sinking Wood**

**Problem Situation 2: Sinking a Block of Wood**

A block of wood, 10cm by 20 cm by 4 cm thick is floated on a pool of water, submerging it to a depth of 1 cm. A 50 gram mass is placed on top of the block, increasing the depth submerged to 1.25. Another 50 gram mass is placed on top, increasing the depth to 1.5.

#5 Points possible: 5. Total attempts: 5

Find a linear model for the depth of the block in terms of the mass placed on top, with *D* representing the depth in centimeters to which the block is submerged, and *w* representing the amount of weight in grams on top of the block.

#6 Points possible: 5. Total attempts: 5

Use the model to predict the mass of the block of wood. (Measure the weight in grams.)

grams

#7 Points possible: 5. Total attempts: 5

How much mass could be added before the block would be totally submerged?

grams

**HW 3.6**

#1 Points possible: 5. Total attempts: 5

A linear model passes through the points (20, 607) and (45, 1182).

Find the equation of the line, with *x* as the input and *y* as the output.

#2 Points possible: 5. Total attempts: 5

t |
16 | 24 | 36 | 40 |

y |
1 | 7 | 16 | 19 |

Given the table of values above, find a linear equation for *y* in terms of *t*.

#3 Points possible: 16. Total attempts: 5

The population of wolves in a sanctuary has been growing by about 20 wolves a year. In 2012, there were 415 wolves in the sanctuary.

a. Create a linear equation for the number of wolves, *P*, using the input variable *n* = number of years since 2012.

b. Create a linear equation for the number of wolves, *P*, using the input variable *t* = number of years since 2010.

c. Predict the number of wolves in the sanctuary in 2018. Try using both equations to ensure they agree. wolves

d. What are the advantages to each way of defining the input variable?

#4 Points possible: 10. Total attempts: 5

It is possible to approximate the outside temperature based on how fast crickets are chirping. Suppose on a 50 degree day, you measure 52 chirps in a minute, and on a 74 degree day, you measure 148 chirps in a minute.

a. Based on this data, determine a linear equation that will output the number of chirps in a minute, *n*, given the temperature in degrees, *t*.

b. Use your equation to predict the temperature if you heard 96 chirps in a minute, to the nearest degree. degrees

#5 Points possible: 10. Total attempts: 5

Match each linear equation with its graph GKRBP

Equation

· y=−3xy=-3x

· y=3y=3

· y=x+2y=x+2

· y=−14xy=-14x

· y=3x+2y=3x+2

Graph Color

a. blue (B)

b. black (K)

c. green (G)

d. red (R)

e. purple (P)

#6 Points possible: 12. Total attempts: 5

RGBK

If all the graphs above have equations with form y=mx+by=mx+b,

a. Which graph has the largest value for *b*?

b. Which graph has the smallest value for *b*?

c. Which graph has the largest value for *m*?

d. Which graph has the smallest value for *m*?

#7 Points possible: 5. Total attempts: 5

A cell phone carrier charges a fixed monthly fee plus a constant rate for each minute used. Part 1. In January, the total cost for 400400 minutes was $82$82 while in February, the total cost for 100100 minutes was $58$58. The constant charge for each minute used is:

· 0.1

· 0.09

· 0.08

Part 2. The fixed montly fee charged by the carrier is; fee = $

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