Suppose that a sum S0 is invested at an annual rate of return r compounded continuously. (a) Find the time T required for the original sum to double in value as a function of r. (b) Determine T if r = 7%. (c) Find the return rate that must be achieved if the initial investment is to double in 8 years. Please explain!

a) T = ln2/ln[(100+r)/100]

b) 10.2 years (3 sf)

c) 9.05% (3 sf)

Step-by-step explanation:

r% return: amount is (100+r)%

S = S0 × [(100+r)/100]^T

S = 2S0

2S0 = S0×[(100+r)/100]^T

2 = [(100+r)/100]^T

ln(2) = T ln[(100+r)/100]

T = ln2/ln[(100+r)/100]

b) r = 7

T = ln2/ln[(100+7)/100]

T = ln2/ln1.07

T = 10.24476837 years

T = 10.2 years (3 sf)

c) T = 8

8 = ln2/ln[(100+r)/100]

ln[(100+r)/100] = ln2/8

1 + r/100 = 1.090507733

r/100 = 0.090507733

r = 9.0507733%

r = 9.05% (3 sf)