High School

Use the following information to answer the next two exercises.

An electronics retailer used regression to find a simple model to predict sales growth in the first quarter of the new year (January through March). The model is good for 90 days, where \( x \) is the day. The model can be written as follows:
\[ \hat{y} = 101.32 + 2.48x \]
where \( \hat{y} \) is in thousands of dollars.

8. What would you predict the sales to be on day 60?

9. What would you predict the sales to be on day 90?

Use the following information to answer the next three exercises.

A landscaping company is hired to mow the grass for several large properties. The total area of the properties combined is 1,345 acres. The rate at which one person can mow is as follows:
\[ \hat{y} = 1350 - 1.2x \]
where \( x \) is the number of hours and \( \hat{y} \) represents the number of acres left to mow.

10. How many acres will be left to mow after 20 hours of work?

11. How many acres will be left to mow after 100 hours of work?

12. How many hours will it take to mow all of the lawns? (When is \( \hat{y} = 0 \)?)

Answer :

1. The predicted sales on day 60 would be $250,120.

2. The predicted sales on day 90 would be $324,520.

3. After 20 hours of work, there will be 1326 acres left to mow.

4. After 100 hours of work, 1230 acres will be left to mow.

5. It will take approximately 1125 hours to mow all of the lawns.

How to predict the sale

To predict the sales on day 60, substitute x = 60 into the given model:

ŷ = 101.32 + 2.48x

ŷ = 101.32 + 2.48(60)

ŷ = 101.32 + 148.8

ŷ ≈ 250.12

Therefore, the predicted sales on day 60 would be approximately $250,120.

Similarly, to predict the sales on day 90, we substitute x = 90 into the model:

ŷ = 101.32 + 2.48x

ŷ = 101.32 + 2.48(90)

ŷ = 101.32 + 223.2

ŷ ≈ 324.52

Hence, the predicted sales on day 90 would be approximately $324,520.

To determine the number of acres left to mow after 20 hours of work, we substitute x = 20 into the given model:

ŷ = 1350 - 1.2x

ŷ = 1350 - 1.2(20)

ŷ = 1350 - 24

ŷ = 1326

Therefore, after 20 hours of work, there will be approximately 1326 acres left to mow.

Similarly, to find the number of acres left after 100 hours of work:

ŷ = 1350 - 1.2x

ŷ = 1350 - 1.2(100)

ŷ = 1350 - 120

ŷ = 1230

After 100 hours of work, approximately 1230 acres will be left to mow.

To determine the number of hours required to mow all of the lawns, we set ŷ (acres left to mow) equal to 0:

ŷ = 1350 - 1.2x

0 = 1350 - 1.2x

1.2x = 1350

x = 1350 / 1.2

x = 1125

Therefore, it will take approximately 1125 hours to mow all of the lawns.

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