Math help!!! Geometric series word problem?

Question

please show work as there are other question like this.

A hard rubber ball is dropped from a moving van with a height of 5m. The ball rises to 80% of the previous height after each bounce.

a) determine the total vertical distance the ball have travelled when it strikes the ground the eighth time.

b) estimate the total vertical distance the ball travels before it comes to rest.

Answer 1

Let  a = 5m and r = 80% = 80/100 = 0.8

Geometric progressions form is a + ar + ar² + ar³ + ................. + arⁿ

a) determine the total vertical distance the ball have traveled when it strikes the ground the eighth time.

So n = 8 -1 = 7

= 5 +2[5(0.8)] +5(0.8)² +.......+5(0.8)⁷]

Take out common term 5.

 

= 5 + 2(5)[(0.8)+(0.8)²+(0.8)³+(0.8)⁴+(0.8)⁵+(0.8)⁶+(0.8)⁷]

 

Simplify

 

=5+10(0.8+0.64+0.512+0.4.96+0.32768+0.262144+0.209715)

 

= 5 + 10(3.220539)

 

= 5 + 32.20539

=37.20539 m

Comments

=5+10(0.8+0.64+0.512+0.4096+0.32768+0.262144+0.209715)

= 5 + 10(3.161139)

= 5 + 31.61139

= 36.6139 m

Total vertical distance = 36.61m.

Answer 2

Let  a = 5m and r = 80% = 80/100 = 0.8

Geometric progressions form is a + ar + ar² + ar³ + ................. + arⁿ

b) estimate the total vertical distance the ball travels before it comes to rest.

= 5 +2[5(0.8)] +5(0.8)² + 5(0.8)³.....................]

Take out common term 5.

 

= 5 + 2(5)[(0.8)+(0.8)²+(0.8)³+.........................]

 

= 5 + (10)[(0.8)+(0.8)²+(0.8)³+.........................] ------------> (1)

 

Consider

 

[(0.8)+(0.8)²+(0.8)³+.........................]

 

Take out common term 0.8.

 

(0.8)[1+(0.8)+(0.8)²+(0.8)³+.........................]

 

Here a= 0.8 and r = 0.8

 

= a / (1 - r )

= 0.8 / (1 - 0.8)

=0.8 / 0.2

=  4

 

substitute =  4 in the equation(1).

 

= 5 + (10)(4)

 

= 5 + 40

 

= 45 m.