1.Suppose you pay $16,000 for a new car and estimate that it will be worth 20% less with each passing year. What will your car be worth after 3 years?
2.A school club uses a telephone chain to inform members of changes in plans. The club president calls 2 members, each of whom calls 2 members, and so on. The process must be repeated 5 times. How many members are in the club?
For 1. i got about 10,000 and 2. i got 32 members
2.
Use the formula for partial sums Sn=a1(1-r^n)/1-2.
a1 is 1 because it started off as one person.
r is 2 (rate of people calling)
n is 6 because the process is repeated 5 times (plus the first call)
so s6=1(1-2^6)/1-2
s6=63.
1. To calculate the value of the car after 3 years, we need to calculate the decreases in value for each year.
Year 1: You estimate a decrease of 20% on the initial value of $16,000.
Value after Year 1 = $16,000 - (20/100) * $16,000 = $16,000 - $3,200 = $12,800
Year 2: You estimate a decrease of 20% on the value after the first year.
Value after Year 2 = $12,800 - (20/100) * $12,800 = $12,800 - $2,560 = $10,240
Year 3: You estimate a decrease of 20% on the value after the second year.
Value after Year 3 = $10,240 - (20/100) * $10,240 = $10,240 - $2,048 = $8,192
Therefore, the car will be worth $8,192 after 3 years.
2. To calculate the number of members in the club after 5 iterations of the telephone chain, we can use exponentiation.
Initial number of members = 1 (club president)
Number of members after the first iteration = 1 + 2 = 3
Number of members after the second iteration = 3 + (2 * 2) = 7
Number of members after the third iteration = 7 + (2 * 2 * 2) = 15
Number of members after the fourth iteration = 15 + (2 * 2 * 2 * 2) = 31
Number of members after the fifth iteration = 31 + (2 * 2 * 2 * 2 * 2) = 63
Therefore, there are 63 members in the club after 5 iterations of the telephone chain.
To calculate the value of the car after 3 years, you need to apply the 20% depreciation rate for each year. Here's how you can find the answer:
1. Start with the initial value of the car, which is $16,000.
2. Apply a 20% depreciation for the first year: $16,000 - (0.20 * $16,000) = $16,000 - $3,200 = $12,800. The car is now worth $12,800 after the first year.
3. Apply another 20% depreciation for the second year: $12,800 - (0.20 * $12,800) = $12,800 - $2,560 = $10,240. The car's value after the second year is $10,240.
4. Apply a 20% depreciation for the third year: $10,240 - (0.20 * $10,240) = $10,240 - $2,048 = $8,192. The car will be worth $8,192 after three years.
So, after 3 years, your car will be worth $8,192.
For the second question, to determine the number of club members based on the telephone chain scenario, you need to count the number of people involved in each round. Here's how you can calculate it:
1. In the first round, the club president calls 2 members.
2. In the second round, each of the 2 members calls 2 members, so there are 2 * 2 = 4 members involved.
3. In the third round, each of the 4 members calls 2 members, so there are 4 * 2 = 8 members involved.
4. In the fourth round, each of the 8 members calls 2 members, so there are 8 * 2 = 16 members involved.
5. In the fifth and final round, each of the 16 members calls 2 members, so there are 16 * 2 = 32 members involved.
To calculate the total number of club members, you need to sum up the number of members involved in each round: 2 (first round) + 4 (second round) + 8 (third round) + 16 (fourth round) + 32 (fifth round) = 62.
Therefore, the club has a total of 62 members.
16,000 (.8)^3 = 8192
2^5 = 32