1. If 100 is saved on October 1, 150 on October 2, 200 on October 3, and so on, how much is saved during October?
2. A Ping-Pong ball is dropped from a height of 16 ft and always rebounds 1/4 of the distance of the previous fall.
a. What distance does it rebound the 6th time?
b. What is the total distance the ball has travelled after this time?
1. To find the total amount saved during October, we can add up the amounts saved each day.
On October 1, $100 is saved.
On October 2, an additional $150 is saved, making a total of $100 + $150 = $250.
On October 3, an additional $200 is saved, making a total of $250 + $200 = $450.
And so on...
To calculate the total amount saved during October, we need to find the sum of an arithmetic sequence. The formula to find the sum of an arithmetic sequence is given by S = (n/2)(a + l), where S is the sum, n is the number of terms, a is the first term, and l is the last term.
In this case, the first term a = 100, and we need to find the last term l. The last term represents the amount saved on the last day of October. Assuming October has 31 days, we can find the last term using the formula l = a + (n - 1)d, where d is the common difference (the amount saved each day) and n is the number of terms (31 in this case).
To find the last term l, we can substitute the values into the formula:
l = 100 + (31 - 1)d
l = 100 + 30d
The common difference is given as 50, since the amount saved increases by $50 each day. Substituting this value, we have:
l = 100 + 30(50)
l = 100 + 1500
l = 1600
Now we have the first term a = 100 and the last term l = 1600. The number of terms n is 31.
Substituting these values into the sum formula, we can find the total amount saved during October:
S = (31/2)(100 + 1600)
S = 15.5 * 1700
S ≈ $26,350
Therefore, the total amount saved during October would be approximately $26,350.
2. Let's calculate the distances for each part of the problem:
a. The distance the ball rebounds on the 6th time:
We are given that the ball rebounds 1/4 of the distance of the previous fall. So, if we start with a drop of 16 ft, the rebound distance can be calculated as:
16 ft * (1/4)^6
Simplifying this expression, we have:
16 ft * (1/4) * (1/4) * (1/4) * (1/4) * (1/4) * (1/4) = 16 ft * (1/4)^6
Calculating this, we get:
16 ft * (1/4)^6 = 16 ft * (1/4096) ≈ 0.0039 ft
Therefore, the distance the ball rebounds on the 6th time is approximately 0.0039 ft.
b. The total distance the ball has traveled after the 6th rebound:
To find the total distance, we need to consider the distances covered during each fall and rebound. The total distance can be calculated by adding up all these distances.
The height of the initial fall is 16 ft, then the ball rebounds 1/4 of the previous fall distance.
So, the distances covered during each fall and rebound can be expressed as:
Fall 1: 16 ft
Rebound 1: 16 ft * (1/4)
Fall 2: 16 ft * (1/4) * (1/4)
Rebound 2: 16 ft * (1/4) * (1/4) * (1/4)
... and so on
To find the total distance after the 6th rebound, we need to sum up these distances:
Total distance = 16 ft + 16 ft * (1/4) + 16 ft * (1/4)^2 + 16 ft * (1/4)^3 + 16 ft * (1/4)^4 + 16 ft * (1/4)^5 + 16 ft * (1/4)^6
We can simplify this expression by using the formula for the sum of a geometric series:
Total distance = 16 ft * (1 - (1/4)^7) / (1 - 1/4)
Calculating this expression, we have:
Total distance ≈ 16 ft * (1 - (1/16384)) / (3/4)
Total distance ≈ 16 ft * (16383/16384) / (3/4)
Total distance ≈ 16 ft * (16383/16384) * (4/3)
Simplifying this, we get:
Total distance ≈ 16 ft * 43794304/65536 ≈ 106666.67 ft
Therefore, the total distance the ball has traveled after the 6th rebound is approximately 106666.67 ft.