Question

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.(2 points)
Square Feet and Cleaning Time

Number of Square Feet: 10, 70, ___________, 210
Time in Minutes: 4, ________, 48, 84

Answer 1

To find the missing values in the table, let's analyze the relationship between the square footage and the cleaning time.

From the data provided:
- For 10 square feet, it takes 4 minutes.
- For 70 square feet, we need to find the cleaning time.
- For an unknown number of square feet, it takes 48 minutes.
- For 210 square feet, it takes 84 minutes.

Now, let's find the rates between the known values.

1. From 10 to 70 square feet (60 square feet increase):
- Cleaning time goes from 4 minutes to an unknown value.

2. From 70 to 210 square feet (140 square feet increase):
- Cleaning time goes from an unknown value to 84 minutes.

Next, we calculate feet per minute based on the given points:

For 10 to 210 square feet, we know:

- From 10 to 210: \( 210 - 10 = 200 \) square feet
- From 4 to 84 minutes: \( 84 - 4 = 80 \) minutes

Now, we can find the average rate of cleaning per square foot across this entire range:

Average rate = \(\frac{80 \text{ minutes}}{200 \text{ square feet}} = 0.4 \text{ minutes per square foot}\)

Using this average to estimate cleaning times for the known ranges:

1. For 70 square feet:
\[
70 \text{ square feet} \times 0.4 \text{ minutes/sq ft} = 28 \text{ minutes}
\]

2. The square footage corresponding to 48 minutes can be calculated:
\[
48 \text{ minutes} = 0.4 \text{ minutes/sq ft} \times x \text{ square feet}
\]
Solving for \( x \):
\[
x = \frac{48 \text{ minutes}}{0.4 \text{ minutes/sq ft}} = 120 \text{ square feet}
\]

So now we can fill in the table:

- Number of Square Feet: 10, 70, 120, 210
- Time in Minutes: 4, 28, 48, 84

The completed table is:

| Number of Square Feet | Time in Minutes |
|-----------------------|------------------|
| 10 | 4 |
| 70 | 28 |
| 120 | 48 |
| 210 | 84 |