To determine the height of the ball at the top of the 6th bounce, we notice from the given heights that the sequence of heights forms a geometric sequence. A geometric sequence is one where each term after the first is found by multiplying the previous term by a constant ratio.
Given the heights:
- 1st bounce: 250 feet
- 2nd bounce: 200 feet
- 3rd bounce: 160 feet
- 4th bounce: 128 feet
- 5th bounce: 102.4 feet
First, we need to determine the common ratio [tex]\( r \)[/tex] of the geometric sequence. The ratio can be found by dividing the height of any bounce by the height of the preceding bounce.
To find the common ratio [tex]\( r \)[/tex]:
[tex]\[ r = \frac{\text{Height at 2nd bounce}}{\text{Height at 1st bounce}} = \frac{200}{250} = 0.8 \][/tex]
So, the common ratio [tex]\( r \)[/tex] is 0.8.
Next, we use the formula for the nth term of a geometric sequence:
[tex]\[ a_n = a \cdot r^{(n-1)} \][/tex]
where:
- [tex]\( a_n \)[/tex] is the height at the nth bounce,
- [tex]\( a \)[/tex] is the initial height (height at the 1st bounce),
- [tex]\( n \)[/tex] is the bounce number,
- [tex]\( r \)[/tex] is the common ratio.
Substituting the known values into the formula to find the height at the 6th bounce:
[tex]\[ a_6 = 250 \cdot (0.8)^{(6-1)} \][/tex]
[tex]\[ a_6 = 250 \cdot (0.8)^5 \][/tex]
Now, calculating [tex]\( (0.8)^5 \)[/tex]:
[tex]\[ (0.8)^5 = 0.32768 \][/tex]
Then, multiplying by the initial height:
[tex]\[ a_6 = 250 \cdot 0.32768 = 81.92 \quad \text{feet} \][/tex]
Therefore, the height of the ball at the top of bounce 6 is approximately 81.92 feet.
