Recall that you invest a principal of P with annual interest rate r compounded n times a year. Then, the balance
after t years under this discrete compounding scheme is as follows.
B(t) = P 1 + r
n
n t.
Typically, n = 12 (monthly reinvestment) or n = 4 (quarterly reinvestment) for most consumer accounts.
Bank A offers an investment CD with an annual interest rate of 5.1% compounded monthly, that of bank B offers has
an annual interest rate 5.2% compounded quarterly, and that of bank C has an annual interest rate of 5.3% compounded annually. Which bank is the best for you?
- Logistic Growth Model
Local scientists discovered the spread of a certain disease in their community. The initial infection is 35 people, and
the number of infected people after t days is estimated as follows.
P(t) = 3850
1 + 109.0 ⅇ-0.04 t
(2.1) Estimate the number of infected people after 100 days, 200 days and 365 days.
(2.2) How many days later will the infected population reach 1,500 people?
- Radioactive Decay Model
Americium-241 is a radioactive substance used in common household smoke detectors. The radioactive decay of Americium-241
can be modeled by the following formula, where Q0 is the initial quantity and the time t is measured in years
Q(t) ≈ Q0 ·
1
2
t / 432.2
(3.1) Suppose that you have 10 milligrams of Americium-241. Estimate the remaining quantity of Americium-241
after (a) 20 years; (b) 200 years; (c) 432.2 years; (d) 864.4 years.
(3.2) How many years later will the remaining quantity be 7.0 milligrams?
- Logarithmic Scaling Model
The Richter-scale measurement of earthquake is defined as follows.
R = log10(I / S) .
Here, S is the intensity of the standard earthquake whose amplitude is 1 micron, or 10-6 m.
The 1906 earthquake in San Francisco had an estimated magnitude of 8.3 in Richter scale. In the same year, a
powerful earthquake occurred on the Columbia-Ecuador border that was four times as intense. What was the Richter
scale of the second earthquake?
Full Answer Section
Steps:
- Calculate the effective annual rates for Banks A and B using the formula: Effective rate = (1 + nominal rate/n)^n - 1
- Bank A: Effective rate = (1 + 5.1%/12)^12 - 1 ≈ 5.32%
- Bank B: Effective rate = (1 + 5.2%/4)^4 - 1 ≈ 5.24%
- Compare effective rates: Bank C (5.3%) offers the highest effective annual return, followed by Bank A (5.32%) and then Bank B (5.24%).
Therefore, Bank C is the best option for maximizing your investment return.
2. Logistic Growth Model:
The provided formula (P(t) = 3850 / (1 + 109.0^-(0.04t))) estimates the number of infected people after t days.
Steps:
- Plug in the number of days (t) to find the estimated infected population (P(t)).
- 100 days: P(100) ≈ 1238 people
- 200 days: P(200) ≈ 2742 people
- 365 days: P(365) ≈ 3782 people (approaching the carrying capacity of 3850)
2.2 Reaching 1500 Infected:
We need to solve the equation P(t) = 1500 for t. Unfortunately, this equation cannot be solved algebraically for t. We can use numerical methods (e.g., graphing calculators or spreadsheet software) to find the approximate solution.
Using a graphing calculator, you will find that the infected population reaches 1500 people after approximately 55 days.
3. Radioactive Decay Model:
The formula (Q(t) ≈ Q0 * (1/2)^(t / 432.2)) represents the remaining quantity (Q(t)) of Americium-241 after t years, given the initial quantity (Q0).
3.1 Remaining Americium-241:
- (a) 20 years: Q(20) ≈ 7.08 milligrams
- (b) 200 years: Q(200) ≈ 2.22 milligrams
- (c) 432.2 years: Q(432.2) ≈ 5 milligrams (half of the initial quantity decays every 432.2 years)
- (d) 864.4 years: Q(864.4) ≈ 1.25 milligrams
3.2 Reaching 7 Milligrams:
We need to solve the equation Q(t) = 7 for t. Similar to problem 2.2, this requires numerical methods.
Using a graphing calculator, you will find that the remaining quantity reaches 7 milligrams after approximately 13.8 years.
4. Logarithmic Scaling Model:
The Richter scale formula (R = log10(I / S)) relates the measured intensity (I) of an earthquake to its Richter magnitude (R).
Given that the 1906 San Francisco earthquake had a magnitude of 8.3 and the second earthquake was four times as intense:
- Intensity of second earthquake (I2) = 4 * Intensity of San Francisco earthquake (I1)
Steps:
- Substitute the known values and solve for the magnitude (R2) of the second earthquake.
- We don't need the actual intensity values (I1 and I2) as they cancel out when taking the logarithm.
- 8.3 = log10(I2 / S)
- R2 = log10(4 * I1 / S) = log10(I2 / S) + log10(4) (Logarithm properties)
- Since log10(S) is constant for both earthquakes, subtracting it from both sides doesn't affect the answer