The end of a spring is pulled as far as it will go and then is released. On the first bounce back it extends 135 cm. On its second bounce back it extends 45 cm. On its third bounce back it extends 15 cm. How many times will the spring bounce back before it extends 5/9 cm?
I have no idea how to solve this problem. Could someone please show me the steps?
Sure! This problem involves a spring that is being pulled and released, causing it to bounce back. We are given the extension of the spring after each bounce.
To solve this problem, we can look for a pattern in the extensions of the spring. Let's analyze the given information:
First bounce: The spring extends 135 cm.
Second bounce: The spring extends 45 cm.
Third bounce: The spring extends 15 cm.
From these values, we can observe that the change in extension between each bounce is decreasing by a factor of 3 each time. This suggests that the extensions form a geometric sequence.
To find the number of times the spring will bounce before it extends 5/9 cm, we need to find the next term(s) in the sequence until we reach an extension of 5/9 cm.
Let's calculate the extensions for the subsequent bounces:
Fourth bounce: The spring extends 15 / 3 = 5 cm.
Fifth bounce: The spring extends 5 / 3 = 5/3 cm.
Sixth bounce: The spring extends (5/3) / 3 = 5/9 cm.
So, the spring will bounce back 6 times before it extends 5/9 cm.
To summarize the steps:
1. Analyze the given extensions and look for a pattern.
2. Determine if the sequence is arithmetic or geometric.
3. Calculate the next term(s) in the sequence until you reach the desired extension.
I hope this explanation helps you understand how to solve the problem! Let me know if you have any further questions.