Question

Hanna is shopping for a new deep freezer and is deciding between two models. One model costs \(\$ 500\) and has an estimated electricity cost of S100/year. A second model that is more energy efficient costs S800 but has an estimated electricity cost of \(\$ 60 /\) year. a) For each freezer, write an equation for the average cost per year as a function of the time, in years. b) Graph the functions for a reasonable domain. c) Identify important characteristics of each graph and explain what they show about the situation. d) How can the graph help Hanna decide which model to choose?

Short answer

For less than 7.5 years, choose the first freezer. For more than 7.5 years, choose the second freezer.

Step-by-step solution

Define Variables

Let’s define the important variables. Let C_1 be the total cost for the first freezer. C_2 be the total cost for the second freezer. t be the time in years.

Write the Cost Equation for the First Freezer

The first freezer cost equation includes the initial cost of the freezer and the annual electricity cost. Initial cost: - 500 Annual cost: - 100t. So, the equation: C_1 = 500 + 100t.

Write the Cost Equation for the Second Freezer

The second freezer cost equation includes the initial cost of the freezer and the annual electricity cost. Initial cost: - 800 Annual cost: - 60t. So, the equation: C_2 = 800 + 60t.

Graph the functions

Graph the functions C_1 = 500 + 100t and C_2 = 800 + 60t on the same set of axes. Use a reasonable domain, for example, between 0 and 10 years. Plot the y-axis (cost function) and x-axis (time in years).

Analyze Characteristics of the Graphs

Identify the points where the graphs intersect and where they reach their respective y-intercepts. The y-intercept for C_1 is 500, and for C_2 is 800. The slopes represent the annual electricity costs: 100 for the first and 60 for the second.

Determine Hanna's Choice

Evaluate the crossover point, which is when 500 + 100t = 800 + 60t. Solve for t. Subtract 60t from both sides to get 40t = 300, so t = 7.5 years. If Hanna keeps the freezer for more than 7.5 years, the second model will be cheaper in the long run.

Key concepts

linear equations

Linear equations form the basis of understanding cost functions. Think of a straight line when you hear 'linear equation'. The formula for a linear equation is usually written as \( y = mx + b \), where \( y \) is the output (total cost, in this case), \( x \) is the input (time in years), \( m \) is the slope (annual cost), and \( b \) is the y-intercept (initial cost).
For Hanna's freezer models, let's break it down:
1. First freezer: \( C_1 = 500 + 100t \)
2. Second freezer: \( C_2 = 800 + 60t \)
These equations tell us how much each freezer costs over \( t \) years. By comparing these equations, Hanna can make an economical choice.

graphing functions

Graphing functions helps visualize the cost over time. We plot the cost functions for both freezers on a graph.
1. The x-axis (horizontal) represents time in years.
2. The y-axis (vertical) shows total cost.
For the first freezer: Starting at \( 500 \) (initial cost) and increasing by \( 100 \) per year.
For the second freezer: Starting at \( 800 \) and increasing by \( 60 \) per year.
By graphing both lines on the same axes, we can see where they intersect. This crossover point is crucial because it indicates when the cheaper initial cost freezer becomes more expensive over time.

cost comparison

Cost comparison is about determining which option is cheaper over a specific period. Hanna needs to compare two things:
1. Initial cost of each freezer.
2. Annual electricity cost.
We analyze the linear equations:
1. First freezer: Higher initial cost but lower annual cost.
2. Second freezer: Higher annual cost but lower initial cost.
The intersection point from the graphs tells us the break-even point. Before this point, one is cheaper. After this point, the other is.
This helps Hanna see that while the second freezer costs more initially, it could save money on electricity in the long run.

break-even analysis

Break-even analysis tells us when the total costs of the two options will be equal. Mathematically, we find the intersection of their cost functions.
Set \( C_1 = C_2 \):
\( 500 + 100t = 800 + 60t \).
Solve for \( t \):
1. Subtract 60t from both sides: \( 40t = 300 \)
2. Divide by 40: \( t = 7.5 \)
This means at 7.5 years, Hanna's spending on both freezers will be the same. If she keeps the freezer longer than 7.5 years, the second freezer becomes cheaper overall. This insight can help Hanna decide based on her long-term usage expectations.