Find the number of hours it would take to reach 50. I know that multiplying a previous value by 0.921 gives you the next value for a whole hour (bottom to top). However, I am really struggling to find a way to calculate the hours for the value 50. How can I go about doing this?

equissupnica7 2022-07-20 Answered
Probability - Geometric Sequence Problem
I have been trying to solve a problem which states the following:
For every hour less than 8, an initial value depreciates by 7.9%. The initial value given is 87. This is what I have calculated so far:
8 hours = 87
7 hours = 80.127
6 hours = 73.797
5 hours = 67.967
4 hours = 62.598
3 hours = 57.652
2 hours = 53.098
1 hours = 48.903
From the above, it can be seen that the problem is a geometric sequence, having a constant ratio of 0.921.
The problem asks to find the number of hours it would take to reach 50. I know that multiplying a previous value by 0.921 gives you the next value for a whole hour (bottom to top). However, I am really struggling to find a way to calculate the hours for the value 50. How can I go about doing this?
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Answers (1)

yatangije62
Answered 2022-07-21 Author has 16 answers
Step 1
After each hour, the value V is multiplied by 0.921 (i.e. losing 7.9%).
So after h hours, the value is the initial value, 87, multiplied by 0.921, h times:
V = 87 0.921 0.921 . . . 0.921 (h occurences of 0.921)
This can be written as V = 87 0.921 h , and the formula works for fractional values of h as well as for whole numbers.
Step 2
Rearrange this formula by taking logarithms of both sides (noting that l o g ( A B ) = l o g ( A ) + l o g ( B ) and l o g ( A C ) = C l o g ( A )
So: l o g ( V ) = l o g ( 87 ) + h l o g ( 0.921 )
You can then substitute in the value V = 50 to solve for h
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