The table below models the cost, $y$, of using a high-efficiency washing machine and a standard washing machine over $x$ number of years.
| Number of Years | High-Efficiency Cost($) | Standard Cost($) |
|---|---|---|
| 1 | 525 | 430 |
| 2 | 550 | 460 |
| 3 | 575 | 490 |
Which equation represents the cost of the high-efficiency washing machine over a given number of years?
Which equation represents the cost of the standard washing machine over a given number of years?
After how many years of use would the washing machines cost the same amount?
Which washing machine would be the more practical purchase if kept for 9 years?
Answer (2)
Determine the equation for the high-efficiency washing machine: y = 25 x + 500 .
Determine the equation for the standard washing machine: y = 30 x + 400 .
Find the number of years when the costs are equal: x = 20 years.
Calculate the cost after 9 years and determine the more practical purchase: The standard machine is cheaper at $670 compared to the high-efficiency machine at 725 , makin g i tt h e m ore p r a c t i c a l c h o i ce . \boxed{\text{Standard Machine}}$
Explanation
Understanding the Problem We are given a table that shows the cost, y , of using a high-efficiency washing machine and a standard washing machine over x number of years. We need to find the equations representing the cost of each washing machine, determine when their costs are equal, and identify the more practical purchase after 9 years.
High-Efficiency Washing Machine Equation First, let's find the equation for the high-efficiency washing machine cost. The cost increases linearly with years. We can find the slope, m 1 , using two points from the table, such as (1, 525) and (2, 550). m 1 = 2 − 1 550 − 525 = 1 25 = 25 The equation is of the form y = m 1 x + b 1 . Using the point (1, 525), we have 525 = 25 ( 1 ) + b 1 Solving for b 1 , we get b 1 = 525 − 25 = 500 Thus, the equation for the high-efficiency washing machine is y = 25 x + 500
Standard Washing Machine Equation Next, let's find the equation for the standard washing machine cost. Similarly, we find the slope, m 2 , using two points from the table, such as (1, 430) and (2, 460). m 2 = 2 − 1 460 − 430 = 1 30 = 30 The equation is of the form y = m 2 x + b 2 . Using the point (1, 430), we have 430 = 30 ( 1 ) + b 2 Solving for b 2 , we get b 2 = 430 − 30 = 400 Thus, the equation for the standard washing machine is y = 30 x + 400
Years Until Costs are Equal Now, let's find the number of years, x , when the costs are equal. We set the two equations equal to each other: 25 x + 500 = 30 x + 400 Solving for x , we get 5 x = 100 x = 5 100 = 20 So, the costs are equal after 20 years.
Cost After 9 Years Let's calculate the cost of each washing machine after 9 years.For the high-efficiency machine: y = 25 ( 9 ) + 500 = 225 + 500 = 725 For the standard machine: y = 30 ( 9 ) + 400 = 270 + 400 = 670
Determining the More Practical Purchase After 9 years, the high-efficiency machine costs $725 and the standard machine costs $670. Therefore, the standard machine is the more practical purchase if kept for 9 years, as it is cheaper.
Examples
Understanding the cost of appliances over time can help you make informed purchasing decisions. This type of analysis is useful when comparing different models or brands of appliances with varying energy efficiency and upfront costs. For example, when deciding between a standard refrigerator and an energy-efficient model, you can project the total cost over several years, including purchase price and electricity consumption, to determine which option offers the best long-term value. This approach ensures you consider both initial investment and ongoing expenses, leading to more cost-effective choices for your household.
The equations for the washing machines are: high-efficiency: y = 25 x + 500 and standard: y = 30 x + 400 . They cost the same after 20 years, and after 9 years, the standard machine costs $670 compared to $725 for the high-efficiency model, making the standard machine the better choice. Thus, the more practical purchase is the standard washing machine.
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