a. Write a formula for Q as a function of t.
b. What is the value of Q when t=10
1. Initial amount 2000; increasing by 5% per year
2. Initial amount 112.8; decreasing by 23.4% per year
3. Initial amount 5; increasing by 100% per year
a. Formula for Q as a function of t:
1. Q(t) = 2000 * (1 + 0.05)^t
2. Q(t) = 112.8 * (1 - 0.234)^t
3. Q(t) = 5 * (1 + 1)^t
b. Calculating the value of Q when t = 10:
1. Q(10) = 2000 * (1 + 0.05)^10
Q(10) ≈ 2000 * (1.05)^10
Q(10) ≈ 2000 * 1.6487212707
Q(10) ≈ 3297.44
2. Q(10) = 112.8 * (1 - 0.234)^10
Q(10) ≈ 112.8 * (0.766)^10
Q(10) ≈ 112.8 * 0.2143588815
Q(10) ≈ 24.19
3. Q(10) = 5 * (1 + 1)^10
Q(10) ≈ 5 * (2)^10
Q(10) ≈ 5 * 1024
Q(10) ≈ 5120
To answer these questions, we need to use the formula for exponential growth or decay:
For exponential growth:
Q = P * (1 + r)^t
For exponential decay:
Q = P * (1 - r)^t
Where:
Q = Final amount
P = Initial amount
r = Growth or decay rate (expressed as a decimal)
t = Time period
a. Write a formula for Q as a function of t.
1. For the first scenario, the initial amount is 2000 and it is increasing by 5% per year. So, the growth rate (r) is 0.05 (5% expressed as a decimal). Therefore, the formula for Q as a function of t is:
Q = 2000 * (1 + 0.05)^t
2. For the second scenario, the initial amount is 112.8 and it is decreasing by 23.4% per year. So, the decay rate (r) is 0.234 (23.4% expressed as a decimal). Therefore, the formula for Q as a function of t is:
Q = 112.8 * (1 - 0.234)^t
3. For the third scenario, the initial amount is 5 and it is increasing by 100% per year. So, the growth rate (r) is 1 (100% expressed as a decimal). Therefore, the formula for Q as a function of t is:
Q = 5 * (1 + 1)^t
b. What is the value of Q when t = 10?
To find the value of Q when t = 10, we substitute the value of t into the respective formula from step a.
For scenario 1:
Q = 2000 * (1 + 0.05)^10
For scenario 2:
Q = 112.8 * (1 - 0.234)^10
For scenario 3:
Q = 5 * (1 + 1)^10
Calculate the values using a calculator or a software, and you will get the respective values of Q for each scenario when t = 10.
1.
Q(t) = 2000*1.05^t
Q(10) = 3257.9
2.
Q(t) = 112.8*1.234^t
Q(10) = 923.55
3.
Q(t) = 5*2^t
Q(10) = 5120