Newton's Method ③ How to Create Equations
Let us organize how to create equations for Newton's method. You will be able to solve any problem in "three steps."
Let's Solve It in 3 Steps
- ① Choose the unit:Clearly decide what "1 unit" represents (for example: the amount one cow eats in one day＝1).
- ② Create two equations:
S＋r×T₁＝amount eaten×T₁ (condition 1)
S＋r×T₂＝amount eaten×T₂ (condition 2) - ③ Find r by subtraction, and then find S.
Example Problem (Think in Reverse: How Many Cows Can Be Released?)
When 10 cows eat the grass in a pasture, it is gone in 30 days; with 20 cows, it is gone in 10 days. How many cows can be kept so that the grass will never disappear?
Solution:
① S＋r×30＝10×30＝300 …㋐
② S＋r×10＝20×10＝200 …㋑
③ ㋐－㋑：r×20＝100 → r＝5
④ ㋑：S＋50＝200 → S＝150
⑤ For the grass not to disappear, "amount eaten by the cows per day ／ amount of grass that grows per day" must satisfy ≦.
⑥ With n cows, n≦r＝5
Answer: If the number of cows is kept to 5 or fewer, the grass will not disappear (with 5 cows, the amount eaten and the amount that grows are exactly balanced; with fewer than 5 cows, the grass continues to increase).
When 10 cows eat the grass in a pasture, it is gone in 30 days; with 20 cows, it is gone in 10 days. How many cows can be kept so that the grass will never disappear?
Solution:
① S＋r×30＝10×30＝300 …㋐
② S＋r×10＝20×10＝200 …㋑
③ ㋐－㋑：r×20＝100 → r＝5
④ ㋑：S＋50＝200 → S＝150
⑤ For the grass not to disappear, "amount eaten by the cows per day ／ amount of grass that grows per day" must satisfy ≦.
⑥ With n cows, n≦r＝5
Answer: If the number of cows is kept to 5 or fewer, the grass will not disappear (with 5 cows, the amount eaten and the amount that grows are exactly balanced; with fewer than 5 cows, the grass continues to increase).
Key Points
- The condition for the grass not to disappear is "amount eaten in one day ≦ amount that grows in one day r".
- To find "how many days until the grass disappears," solve S＋r×N＝amount eaten×N.
- To find "how many cows will make the grass disappear in N days," set up an equation with n cows and find n.
- Make a habit of checking what you are being asked to find before creating an equation.
Practice Problems
- In a pasture, the grass disappears in 24 days with 12 cows and in 48 days with 8 cows. How many cows can be kept so that the grass will not disappear?
- A spring initially contains S units of water, and r units spring up every minute. It becomes empty in 20 minutes with 4 pumps and in 10 minutes with 6 pumps. If the amount pumped by one pump per minute is 1, find both S and r.
- A queue grows by 3 people every minute and initially has 90 people. If 8 service counters are opened, how many minutes will it take for the queue to disappear? (One counter serves 5 people per minute.)
Answers and Explanations
- Answer:The number of cows should be 4 or fewer
Explanation: Let the amount one cow eats in one day be 1. Condition ①12 cows×24 days＝288 (total consumption), condition ②8 cows×48 days＝384 (total consumption). S＋r×24＝288 …①, S＋r×48＝384 …②. ②－①：r×24＝96→r＝4. From ①, S＝288－96＝192. Condition for the grass not to disappear: number of cows n≦r＝4. Answer: The number of cows should be 4 or fewer (when n＝4, consumption＝increase, so the amount of grass remains constant). - Answer:S＋r＝60
Explanation: Let the amount one pump draws in one minute be 1. Condition ①4 pumps×20 minutes＝80 (total amount pumped out), condition ②6 pumps×10 minutes＝60 (total amount pumped out). S＋r×20＝80 …①, S＋r×10＝60 …②. ①－②：r×10＝20→r＝2. From ②, S＋20＝60→S＝40. Therefore S＋r＝40＋2＝42. (The intended purpose of the problem: practice finding "S＋r" first.) - Answer:10 minutes
Explanation: One counter serves 5 people per minute. Initial amount S＝90 people, increasing amount r＝3 people/minute, and 8 counters serve 8×5＝40 people per minute. The queue disappears after N minutes: 90＋3N＝40N. 90＝37N. N＝90÷37≈2.43 minutes (approximately 2 minutes 26 seconds). Check: 40－3＝37 people/minute (the effective decrease). The queue disappears in approximately 2.43 minutes. For the question "how many minutes," the answer is approximately 2.4 minutes (or it is finished during the third minute).