A home cleaning service takes more time to clean larger spaces. The table shows the number of minutes it takes for the service to clean a certain number of square feet within a home.
| Number of Square Feet, $x$ | 50 | 80 | 100 | 150 |
|---|---|---|---|---|
| Time in Minutes, $y$ | 20 | 32 | 40 | 60 |
Based on the table, how many minutes would it take for the service to clean 200 square feet?
A. 20 minutes
B. 80 minutes
C. 125 minutes
D. 400 minutes
Answer (2)
Calculate the cleaning rate for each data point: The rate is constant at 0.4 minutes per square foot.
Establish the linear relationship: y = 0.4 x , where y is the time in minutes and x is the number of square feet.
Substitute x = 200 into the equation: y = 0.4 × 200 .
Calculate the cleaning time: The time it would take to clean 200 square feet is 80 minutes.
Explanation
Understanding the Problem We are given a table that shows the time it takes a cleaning service to clean different square footage. We need to determine how long it would take to clean 200 square feet.
Calculating Cleaning Rates First, let's calculate the cleaning rate (minutes per square foot) for each data point in the table. This will help us determine if the relationship between square footage and cleaning time is linear.
Determining the Rate The cleaning rates are:
For 50 square feet: 20 minutes /50 sq ft = 0.4 minutes/sq ft
For 80 square feet: 32 minutes /80 sq ft = 0.4 minutes/sq ft
For 100 square feet: 40 minutes /100 sq ft = 0.4 minutes/sq ft
For 150 square feet: 60 minutes /150 sq ft = 0.4 minutes/sq ft
Establishing a Linear Relationship Since the cleaning rate is constant at 0.4 minutes per square foot, the relationship between the number of square feet and the cleaning time is linear. We can express this relationship as:
y = 0.4 x , where y is the time in minutes and x is the number of square feet.
Calculating the Cleaning Time for 200 Square Feet Now, we can substitute 200 square feet into the equation to find the corresponding cleaning time:
y = 0.4 × 200 = 80
Final Answer Therefore, it would take 80 minutes for the cleaning service to clean 200 square feet.
Examples
Understanding proportional relationships is useful in many real-life situations. For example, if you know the price of one item, you can calculate the price of multiple items if the price is directly proportional to the number of items. Similarly, if you know how much time it takes to travel a certain distance at a constant speed, you can calculate the time it would take to travel a different distance.
To clean 200 square feet, it will take 80 minutes based on the cleaning rate of 0.4 minutes per square foot. This relationship is linear, as confirmed by the calculations from the provided data. The chosen answer is B: 80 minutes.
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