1) For tax purposes a manufacturer is allowed to depreciate the value of a machine by 10% each year. If the original value of the machine was $10,000, what would be its value for tax purposes at the end of 6 years. (Assume the depreciation is computed at the end of each year.)
2) A ball is dropped from a height of 30 feet and rebounds 2/3 of the distance it falls on each bounce. How far has the ball traveled vertically when it hits the ground the 4th time? How far will it travel vertically before coming to a halt? (The convergent value of the infinite sums.)
10000 * 0.9^6
On the nth bounce it has to go a round trip, of height
30*(2/3)^n,
So we want the 1st 4 terms of a G.P. with
a = 30
r = 2/3
30 + 2*S4 = 30+2*30(1 - (2/3)^4)/(1-2/3) = 174.44 ft
174.44 feet😊
1) To find the value of the machine for tax purposes at the end of 6 years, we need to calculate the depreciation for each year and subtract it from the original value.
Given:
Original value of the machine = $10,000
Depreciation rate per year = 10%
To calculate the value of the machine after each year, we repeatedly multiply the value by (1 - depreciation rate).
After 1st year: $10,000 * (1 - 0.10) = $9,000
After 2nd year: $9,000 * (1 - 0.10) = $8,100
After 3rd year: $8,100 * (1 - 0.10) = $7,290
After 4th year: $7,290 * (1 - 0.10) = $6,561
After 5th year: $6,561 * (1 - 0.10) = $5,905
After 6th year: $5,905 * (1 - 0.10) = $5,314.50
Therefore, the value of the machine for tax purposes at the end of 6 years would be $5,314.50.
2) To find the distance traveled vertically when the ball hits the ground for the 4th time, we need to calculate the sum of the geometric series formed by the heights of each bounce.
Given:
Initial height = 30 feet
Rebound distance = 2/3 of the previous height
The sum of a geometric series can be calculated using the formula:
Sum = a * (1 - r^n) / (1 - r)
Where:
a = initial term (30 feet in this case)
r = common ratio (2/3 in this case)
n = number of terms (4 bounces in this case)
Sum = 30 * (1 - (2/3)^4) / (1 - 2/3)
Calculating this using the formula, we get:
Sum = 30 * (1 - 16/81) / (1/3) = 30 * (65/81) / (1/3) = 30 * (65/27) ≈ 72.22 feet
Therefore, the ball has traveled approximately 72.22 feet vertically when it hits the ground the 4th time.
To find the distance the ball will travel vertically before coming to a halt, we need to calculate the sum of an infinite geometric series.
Using the same formula as before, but now with n tending to infinity:
Sum = a / (1 - r)
Sum = 30 / (1 - 2/3) = 30 / (1/3) = 30 * (3/1) = 90 feet
Therefore, the ball will travel 90 feet vertically before coming to a halt.