Section 1.2 Estimation

1. Section 1.2Estimation

2. What You Will Learn • Estimation

3. Estimation • The process of arriving at an approximate answer to a question is called estimation. • Estimates are not meant to give exact values for answers, but are a means of determining whether your answer is reasonable. • We often round numbers to estimate, or approximate, an answer. • The symbol ≈ means is approximately equal to.

4. Example 1: Estimating the Cost of Bagels Sandy Mendez decides to purchase bagels for her colleagues. Estimate her cost if she purchases 29 bagels at $1.99 each.

5. Example 1: Estimating the Cost of Bagels Solution: Round the amounts to obtain an estimate Number Number Rounded 29 30 × $1.99× $2.00 $60.00 The 29 bagels will cost approximately $60.00, written ≈ $60.00

6. Example 4: Using Estimates in Calculations The Martellos traveled 1276 miles from their home in Springfield, Missouri, to Yellowstone National Park. a) If their trip took 19.9 hours of driving time, estimate the average speed in miles per hour, that their car traveled.

7. Example 4: Using Estimates in Calculations Solution: To estimate the average miles per hour, divide the number of miles by the number of hours. Numbers Rounded The car averaged about 65 mph.

8. Example 4: Using Estimates in Calculations The Martellos traveled 1276 miles from their home in Springfield, Missouri, to Yellowstone National Park. b) If their car used 52 gallons of gasoline, estimate the average gas mileage of their car, in miles per gallon.

9. Example 4: Using Estimates in Calculations Solution: To estimate the average gas mileage, divide the number of miles by the number of gallons. Numbers Rounded Car averaged about 26 mpg.

10. Example 4: Using Estimates in Calculations The Martellos traveled 1276 miles from their home in Springfield, Missouri, to Yellowstone National Park. c) If the cost of gasoline averaged $3.51 per gallon, estimate the total cost of the gasoline used.

11. Example 4: Using Estimates in Calculations Solution: To estimate the total cost of gasoline, multiply the price per gallon by the number of gallons. Numbers Rounded 52 × $3.51 50 × $3.50 = $175 The cost of gasoline is about $175

12. Estimates on a Map • Sometimes when working with measurements on a map, it may be difficult to get an accurate estimate because of the curves on the map. • To get a more accurate estimate, you may want to use a piece of string and tape or pins to mark the ends.

13. Example 5: Estimates on a Map In the mid-1800s, thousands of settlers followed trails to the West to gain cheap, fertile land and a chance to make a fortune. One of the main trails to the West was the Oregon Trail, which ran from Independence, Missouri, to the Oregon Territory. Another trail, the California Trail, ran from Fort Hall on the Oregon Trail to Sacramento, California. A map of the Oregon Trail and the California Trail is shown on the next slide.

14. Example 5: Estimates on a Map

15. Example 5: Estimates on a Map a) Using the Oregon Trail, estimate the distance from Independence, Missouri, to Fort Hall in the Oregon Territory.

16. Example 5: Estimates on a Map

17. Example 5: Estimates on a Map Solution: a) Approximately ¾-in represents 250 mi Mark off ¾-in intervals, from Independence, Missouri, to Fort Hall in the Oregon Territory, there are about 4, thus the distance is 4 × 250 mi or about 1000 mi

18. Example 5: Estimates on a Map b) Using the California Trail, estimate the distance from Fort Hall in the Oregon Territory to Sacramento, California.

19. Example 5: Estimates on a Map

20. Example 5: Estimates on a Map Solution: On a map like this one, it may be difficult to get an accurate estimate because of the curves on the map. To get an accurate estimate, one can use a string placed along the road, then stretch the string, mark off ¾-in intervals or measure string, about 3¼ in or 1083 mi

21. Example 5: Estimates on a Map Solution: b) Using the procedure discussed in part (a), we estimate that the distance from Fort Hall in the Oregon Territory to Sacramento, California is about 1 5/8 in or about 2.2 ¾-in intervals The distance is about 2.2 × 250 mi, or about 550 mi

22. Estimates on a Photo • In order to estimate large areas, a photograph can be helpful. We can divide the photo into rectangles with equal areas, then pick one area that looks representative of all the areas. • Estimate (count) the number of items in this single area. Then multiply by the number of equal areas.

23. Estimates on a Photo • The number of regions (rectangles) into which you choose to divide the photo is arbitrary. Generally, the more regions, the better the approximation.

24. Example 7: Estimates on a Photo Estimate the number of corn kernels in the top layer in the accompanying photograph.

25. Example 7: Estimates on a Photo Solution: Divide into 20 approximately equal areas, select the top left as representative region, enlarge the area and count the kernels

26. Example 7: Estimates on a Photo Solution: If half a kernel is in the region, we count it. There are about 18 kernels in this region. Multiplying by 20 gives18 × 20 = 360. Thus, there are about 360 kernels in the photo.

27. Homework P. 14 # 3 – 54 (x3)