Algebra 2 by Richard Wright

Previous LessonTable of Contents Next Lesson

Are you not my student and
has this helped you?

0-02 Use Problem Solving Strategies and Models

The United States uses the Fahrenheit temperature scale. Most of the world uses Celsius. This means that temperatures need to be converted from one temperature scale to the other using a formula.

Some Formulas

Strategies to Solve Real-Life (Word) Problems

Often it is easiest to write an equation in words before you write it in mathematical equations. This is called a verbal model. You probably think this way in your head already when you are trying to solve problems.

Example 1

In those days Caesar Augustus issued a decree that a census should be taken of the entire Roman world. … And everyone went to their own town to register. So Joseph also went up from the town of Nazareth in Galilee to Judea, to Bethlehem the town of David, because he belonged to the house and line of David. Luke 2:1-4

It is 68.9 miles from Nazareth to Bethlehem in a straight line, but Joseph and Mary likely went in a route that was about 90 miles. If the average walking speed is 3 miles per hour, how many hours did Joseph and a pregnant Mary have walk?

Solution

Strategy: Use a formula. Since the problem contains distance and speed, which is a rate, use d = rt.

d = rt

$$ 90 \text{ mi} = \left(3\frac{\text{mi}}{\text{hr}}\right)t $$

30 hr = t

Taking time to rest, it has been estimated that the trip took about 9 days.

Example 2

The following table shows the total number of birds, b, seen at a bird feeder for several hours, h, during a day. Find the total number of birds after 7 hours.

Time (hours), h	1	2	3	4	5
Birds, b	4	8	12	16	20

Solution

Strategy: Look for a pattern. During every hour the number of birds increases by 4. Add 4 two more times. Total number of birds after 7 hours is 20 + 2×4 = 28 birds.

Example 3

On a business trip, a Hyundai Elantra car used 18 gallons of gasoline and traveled a total distance of 585 miles. The car's fuel efficiency is 34 miles per gallon on the highway and 25 miles per gallon in the city. How many gallons of gasoline were used in the highway?

Solution

Strategy: Use a formula. Since the problem is about rates, the rate formula should work. d = rt where d = distance, r = fuel rate, t = gallons of gasoline. There are two rates, so use two rt's.

The amount of gas used on the highway will be x. Since the total amount of gallons used is 18 gal, the amount used in the city will be the difference, 18 gal − x.

$$ \begin{array}{rll} d & =r_{highway}t_{highway}+r_{city}t_{city} & \\ 585 \text{ mi} & =\left(34\frac{\text{mi}}{\text{gal}}\right)x+\left(25\frac{\text{mi}}{\text{gal}}\right)\left(18 \text{ gal}-x\right) & \\ 585 \text{ mi} & =\left(34\frac{\text{mi}}{\text{gal}}\right)x+450 \text{ mi}-\left(25\frac{\text{mi}}{\text{gal}}\right)x & \gets\mathrm{Distribute} \\ 585 \text{ mi} & =\left(9\frac{\text{mi}}{\text{gal}}\right)x+450 \text{ mi} & \gets\mathrm{Combine\ like\ terms} \\ 135 \text{ mi} & =\left(9\frac{\text{mi}}{\text{gal}}\right)x & \gets\mathrm{Subtract}\ 450\ \mathrm{mi\ from\ both\ sides} \\ 15 \text{ gal} & =x & \gets\mathrm{Divide\ both\ sides\ by\ }9\frac{\text{mi}}{\text{gal}} \end{array} $$

The car used 15 gallons of gas on the highway.

Practice Exercises

Use the given formula to solve for the missing variable.

1. Distance/rate formula, d = 15 km, r =  ? , t = 3 h
2. Circumference of a circle, C = 10π, r =  ? 
3. Area of a triangle, A = 24 in.², b = 6 in., h =  ? 
4. Temperature, F = 68 °F, C =  ? 

Use a formula to solve the following problems. Write the formula before you solve the problem.

5. You are creating a circular garden 15 feet across. You will use begonias to create a colorful design. Four begonia plants can be planted per square foot. How many begonia plants do you need to complete your garden?
6. During a physics lab, a formula requires the temperature in degrees Celsius, but all you have is a Fahrenheit thermometer which reads 185 °F. What is that temperature in Celsius?
7. Suzy participated in a 10 km run-walk. She walked at 5 km/h and ran at 8 km/h. If she took a total of 1.5 hours, how much time did she spend walking and running?

Find a pattern to solve the following problems. Describe the pattern.

8. A hang-glider's height data is given in the table. How high will the hang-glider be after 7 minutes?
   

   Time (min)	0	1	2	3	4
   Height (ft)	300	277	254	231	208

9. Water is evaporating from your small kiddie pool in your backyard. You measure the depth of the water every day after school. On which day will the pool be empty?
   

   Time (day)	1	2	3	4
   Depth (in.)	8	6.4	4.8	3.2

10. Exercising is typically good for you, so Abby and Ben have challenged each other to see who can get the most steps every day. The table shows their number of steps for the first several days. If this trend continues, on which day will Abby have twice as many steps as Ben?
    

    Time (day)	1	2	3
    Abby's Steps	4020	4032	4044
    Ben's Steps	4010	3812	3614

Draw a diagram to solve the following problems. Draw the diagram.

11. Frank is designing a stained-glass window. The outside frame is a square. He connects the midpoints of each side to form another square. If the length of a side of the outside frame is 10 inches, how long is a side of the inner square to the nearest tenth of an inch?
12. Four friends are playing tug-o-war. Angela pulls to the left with 120 N of force. Bobbie pulls right with 130 N. Charlie also pulls right at 185 N, but Daniel tugs to the left with 190 N. Which team wins and by how much force do they win?

State which problem-solving strategy is best for each problem. Then solve the problem.

13. Jesus said, “Therefore go and make disciples of all nations… And surely I am with you always, to the very end of the age.” Matthew 28:19-20.
    If you make 2 disciples for Jesus and they each make 2 more disciples, the data for several generations is given in the table. How many generations until there are over 100 disciples?

    Generations	1	2	3	4	5
    Disciples	2	4	8	16	32

14. Water drains out of the St. Joseph river at about 3200 ft³/s. How much water flows out of the river in 1 day?
15. Sally is making a picture collage. Part of it will have 4 evenly spaced pictures spread over 24 inches. Their widths are 3 in., 4 in., 5 in., and 5 in. She also wants 2 inches between the edges of the collage and the group of pictures. How far apart should Sally place the pictures?

Mixed Review

16. (0-01) Solve. 2x + 1 = 17
17. (0-01) Solve. 3(x − 1) < 12
18. (0-01) Solve. −x − 1 > 13
19. (0-01) Solve. 2(x + 1) = 5(x − 4)
20. (0-01) Solve for y. 2x + 5y = 10

Answers

1. 5 km/h
2. 5
3. 8 in.
4. 20 °C
5. 707 plants
6. 85 °C
7. 2/3 hr walking, 5/6 hr running
8. 139 ft
9. 6th day
10. 10th day
11. 7.1 in.
12. Bobbie and Charlie win by 5 N
13. 7 generations
14. 276,480,000 ft³ of water
15. 1 in. apart
16. 8
17. x < 5
18. x < −14
19. \(\frac{22}{3}\)
20. \(y=-\frac{2}{5}x+2\)