Question

I need help with part b, I have included what ive gotten so far but I am not sure how to present the equation for P in the way he is asking.

Problem 7: Betty and Bob have lost some data for an account that earns interest at a fixed continuous rate per annum, R. a. After 3 years and 6 months the account value is $5,600. After 8 years and 9 months the account value is $12,300. Algebraically find the rate R, and the initial investment, P. Your final answers should be correct to 3 places after the decimal point. b. Algebraically generalize your work. After 7 years the account value was A. After T, years the account value was A. Find the continuous per annum rate, R, and the initial investment, P. Be sure to show all of your algebraic work. Take care to simplify the expression for P into a quotient with a nice power of A, in the numerator and a nice power of A in the denominator. The powers (exponents) should only involve T, and T,. E should not appear. The section of your text on exponents and logarithms should be helpful

Answer 1

a) Let the principal be P and r be the rate of interest continuously compounded

So, P*e^(3.5*r) = 5600

P*e^(8.75*r) = 12300

Dividing the second equation by the first

=> e^(8.75*r - 3.5*r) = 12300/5600 = 2.196429

=> 5.25 *r = ln (2.196429) = 0.786833

=> r = 0.149873 = 14.987%

P* e^(3.5*0.149873) = 5600

=> P = 5600 /1.6897 = $ 3314.1842

b) In general, if A1 is the amount after T1 years and A2 is the amount after T2 years

P*e^(T1*r) = A1......................................................(1)

P*e^(T2*r) = A2......................................................(2)

Dividing the second equation by the first

e^ (T2*r-T1*r) = A2/A1

taking natural log of both sides

r* (T2-T1) = ln (A2/A1)

=> r = ln(A2/A1) / (T2-T1)

Putting this value in the first and 2nd equation

P* e^(T1/(T2-T1)*ln(A2/A1)) = A1

taking natural log of both sides

ln (P) + (T1/(T2-T1)*ln(A2/A1) = ln(A1)

=> ln (P) + T1/(T2-T1) * ln(A2) - T1/(T2-T1) * ln (A1) = ln (A1)

=>ln(P) + T1/(T2-T1) * ln(A2) = ln (A1)*T2/(T2-T1)  

=> ln(P) = {T2 * ln(A1) - T1 * ln (A2) } / (T2 -T1)   

=> ln (P) = {ln(A1)^T2/(T2-T1) - ln(A2)^T1/(T2-T1)}

=> ln (P) =  ln(A2)^T1/(T1-T2) - ln (ln(A1)^T2/(T1-T2)

=> ln (P) = ln (A2^T1/(T1-T2) / A1^T2/(T1-T2) )

So, P = A2^T1/(T1-T2) / A1^T2/(T1-T2)

which is the required expression