Math Problem Statement
Solution
Let's break down and solve the given problem step by step.
Problem Summary:
- Abhi and Gita start from points P and Q, which are 81 km apart.
- Abhi walks at a constant speed of 4 km/h.
- Gita starts at the same speed (4 km/h) but increases her speed by 0.25 km/h at the end of each hour.
We need to find out how many hours will Abhi take to reach point Q after meeting Gita.
Step-by-Step Solution:
Let:
- be the time in hours when Abhi and Gita meet.
- be the total distance between them (81 km).
-
Abhi's distance: Since Abhi walks at a constant speed of 4 km/h, the distance he covers in hours is:
-
Gita's distance: Gita's speed increases every hour. Let's calculate her total distance after hours.
- In the first hour, her speed is 4 km/h.
- In the second hour, her speed is 4.25 km/h.
- In the third hour, her speed is 4.5 km/h, and so on.
Therefore, the distance Gita covers can be written as the sum of an arithmetic progression where:
- First term () = 4 km/h,
- Common difference () = 0.25 km/h,
- Number of terms = .
The distance Gita covers in hours is the sum of the first terms of the arithmetic progression:
-
Total distance covered by both Abhi and Gita must add up to 81 km:
Substituting the expressions for their distances, we get:
Now, let's solve this equation for . I'll calculate the value of .The solution shows that Abhi and Gita meet after 9 hours.
Next, we need to calculate the total time Abhi takes to reach point Q after meeting Gita.
- Abhi's total time to reach Q: After meeting Gita, Abhi has already walked for 9 hours. Since the total distance from P to Q is 81 km, and Abhi walks at 4 km/h, his total time to cover the entire distance is:
Since Abhi has already walked for 9 hours, the remaining time after meeting Gita is:
Thus, it will take Abhi 11.25 hours to reach point Q after meeting Gita.
Final Answer: The correct option is (2) 11.25 hours.
Would you like further details or have any questions? Here are 5 related questions:
- How would the solution change if Gita increased her speed by 0.5 km/h every hour?
- What if Abhi’s speed was reduced to 3 km/h?
- Can this problem be solved using graphs? How would you plot the speeds?
- What would happen if the total distance was doubled but speeds stayed the same?
- How would the time change if Gita increased her speed every 30 minutes instead of every hour?
Tip: When solving problems with varying speeds, arithmetic progressions are a useful tool to sum up distances effectively.
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Math Problem Analysis
Mathematical Concepts
Algebra
Arithmetic Progressions
Formulas
Distance by Abhi = 4 * t
Distance by Gita = t * (4 + (t-1) * 0.25 / 2)
Total distance = Distance by Abhi + Distance by Gita
Theorems
Sum of Arithmetic Progression
Suitable Grade Level
Grades 9-11
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