Question

) If Daniella opens a mutual fund account with a deposit of $15,000 and it promises to

pay an annual interest rate of r%, compounded on a monthly basis.
(i) Derive a function B, which shows the balance on Daniella′
s account the end of 6 years.
(ii) Make a comparison of the rates of change in B with resect to the interest rate when r is
8% , 10% and 12%.
(b) A virus is believed to be spreading through a sheep farm according to that equation
N = 450 (
1
1 + 249e
−0.1t ) where
N is the number of sheep infected and t is the number of days.
(i) How many sheep are infected at t = 0?
(ii) Using a spreadsheet calculate the number of sheep infected after 0, 10, 30 , 50, 70, 100 days)
(iii) Will all the sheep become infected eventually? Give reasons to support your answer.

Answer 1

Explanation:

(a)(i) The formula for compounded interest is:

A = P (1 + r/n)^(nt)

where A is the final amount,

P is the initial amount,

r is the annual interest rate,

n is the number of compoundings per year,

and t is the number of years.

In this case, P = 15000, n = 12, and t = 6.

B = 15000 (1 + r/12)⁷²

(a)(ii) Take derivative with respect to r using power rule and chain rule.

dB/dr = 90,000 (1 + r/12)⁷¹

Evaluate at different values of r:

`\left[\begin{array}{cc}r&dB/dr\\0.08&144253.51\\0.10&162231.56\\0.12&182414.79\end{array}\right]`

(b)(i) N(t) = 450 [1 / (1 + 249e^(-0.1t))]

N(0) = 450 [1 / (1 + 249)]

N(0) = 1.8

(b)(ii) Evaluate N at different values of t.

`\left[\begin{array}{cc}t&N\\0&1.8\\10&4.9\\30&33.6\\50&168.1\\70&366.7\\100&445.0\end{array}\right]`

(b)(iii) As t approaches infinity, N approaches 450. So yes, eventually all 450 sheep will be infected.