Maths word problem

Question

A sculpture is 180cm tall. The sculpture is made of ten sections stacked on top of each other. Each section is shorter than the one blelow it. The sections get shorter at an even rate. The top section is 5 cm tall. What percentage is each section shorter by compared to the one below it?

Answer 1

I think the answer is that each successive higher section is about 78.99%(.7898944968) shorter than the one right below it. 

 

I set set up the equation 5[1+ (1+x) + (1+x)² + (1+x)³ +... + (1+x)⁸ + (1+x)⁹] = 180.

 

(To represent the sum of the lengths of each of the 10 levels equals 180.)

 

Solving for x, x = .265992

 

1+x = 1.265992, which means that if the top level is length 5 (given), the one right below is 5(1.265992) = 6.32996

 

The level right below that is 5(1.27151108)² = 8.01367872

 

Etc.  

 

Since we know that each section is shorter than the one directly below it by the same rate (given), it must be true that LevelA/LevelB = LevelB/LevelC = LevelC/LevelD = etc. 

 

in in other words, 5/6.32996= 6.32996/8.01367872 (etc) = ... =

 

.7898944069= 78.99%

 

This may not be the easiest way to solve this problem, but it is the one that came to mind. Hope that you understand!

 

You may also realize that the rate of increase as you go down each level 1.265992 is the reciprocal of the rate of decrease (.7898944069) as you go up each level. Do you see why? 

 

1.265992^-1 = .7898944069, and vice versa.