Penelope needs to borrow $11,000. She can borrow the money at 5.5% simple interest for 4 yr or she can borrow at 4% with interest compounded continuously for 4 yr.
(a) How much total interest would Penelope pay at 5.5% simple interest?
(b) How much total interest would Penelope pay at 4% interest compounded continuously?
(c) Which option results in less total interest?
To calculate the total interest for each option, we can use the formula for simple interest and compound interest:
(a) For 5.5% simple interest:
Simple Interest = Principal x Rate x Time
Simple Interest = $11,000 x 0.055 x 4 years
Simple Interest = $11,000 x 0.22
Simple Interest = $2,420
Therefore, Penelope would pay $2,420 in total interest at 5.5% simple interest.
(b) For 4% interest compounded continuously:
Compound Interest = Principal x e^(rt) - Principal
Compound Interest = $11,000 x (2.71828)^[(0.04) x 4] - $11,000
Compound Interest = $11,000 x (2.71828)^(0.16) - $11,000
Compound Interest = $11,000 x 1.16509 - $11,000
Compound Interest = $1,815.99
Therefore, Penelope would pay $1,815.99 in total interest at 4% interest compounded continuously.
(c) Comparing the two options, Penelope would pay less total interest at 4% interest compounded continuously ($1,815.99) compared to 5.5% simple interest ($2,420). Therefore, borrowing at 4% interest compounded continuously would result in less total interest.