TMM Problem 2.3

​Thinking with Mathematical Models

Problem 2.3: Equations for Linear Functions

​TMM Problem 2.3

​Tree Top Fun

Tree Top Fun (TTF, for short) runs adventure sites with zip lines, swings, rope ladders, bridges, and trapezes. The company uses mathematical models to relate the number of customers, prices, costs, income, and profit at its many locations.

​TMM Problem 2.3

To write an equation for a situation...

• Use the form y = mx + b to represent a linear relationship.

• M represents your slope (or rate of change). Ask yourself "As one variable increases, what value is being constantly added (or subtracted)?"

• B represents your y-intercept (or initial value). Ask yourself "What value do we begin with yet does NOT get added on (or subtracted by) again?"

Multiple Select

What does b represent in the equation y = mx + b?

(Select all that apply.)

• initial value

• rate of change

• slope

• y-intercept

Multiple Select

What does m represent in the equation y = mx + b?

(Select all that apply.)

• initial value

• rate of change

• slope

• y-intercept

​TMM Problem 2.3

We will be looking at expenses, income, and profit.

• ​Expenses for a business are money going out. For instance, an expense for TTF would be the cost of the bridges that they have to build.

• Income is money coming in for a business. For instance, income for TTF would be the amount of money a customer pays to ride on a zip line.

• Profit is the amount of money a business has after subtracting the expenses from the income.

Multiple Choice

You're writing an equation that relates the daily income i to the number of customers n.

The standard charge per customer at TTF is $25. Does this represent the rate of change or the initial value?

• rate of change

• initial value

Multiple Choice

Write an equation that relates the daily income i to the number of customers n:

The standard charge per customer at TTF is $25.

• i = 25n

• i = n + 25

• n = 25i

• n = i + 25

Multiple Choice

You're writing an equation that relates daily profit P to the number of customers n.

Each TTF site has daily operating costs of $500. Does this represent the rate of change or the initial value?

• rate of change

• initial value

Multiple Choice

Write an equation that relates daily profit P to the number of customers n:

Each TTF site has daily operating costs of $500. The standard charge per customer at TTF is $25.

• P = 25n + 500

• P = 25n - 500

• n = 25P + 500

• n = 25P - 500

Multiple Choice

You're writing an equation that relates the unpaid balance B after m monthly payments.

One TTF site bought a new rope bridge for $4,500. TTF will make monthly payments of $350 until the bill is paid. Does $350 represent the rate of change or the initial value?

• rate of change

• initial value

Multiple Choice

Write an equation that relates the unpaid balance B after m monthly payments.

One TTF site bought a new rope bridge for $4,500. TTF will make monthly payments of $350 until the bill is paid.

• m = 4500 - 350B

• m = 4500 + 350B

• B = 4500 - 350m

• B = 4500 + 350m