Question

Bob and Alice want to remodel their bathroom in 4 years. They estimate the job will cost $35,000. How much must they invest now at an annual interest rate of 4% compounded quarterly to achieve their goal? The Morenos invest $9000 in an account that grows to $11,000 in 4 years. What is the annual interest rate r if interest is compounded
a. Quarterly
b. Continuously
A) a. 2.192% b.2.179%
B) a. 4.5432% b.4.5153%
C) a. 6.0576% b. 6.0204%
D) a. 5.048% b.5.017%

Answer 1

Answer:

Bob and Alice want to remodel their bathroom in 4 years. They estimate the job will cost $35,000. How much must they invest now at an annual interest rate of 4% compounded quarterly to achieve their goal?

we need to use the present value formula:

present value = future value / (1 + interest rate)ⁿ = $35,000 / (1 + 1%)¹⁶ = $29,848.74

The Morenos invest $9000 in an account that grows to $11,000 in 4 years. What is the annual interest rate r if interest is compounded

a. Quarterly

$9,000 = $11,000 / (1 + r)¹⁶

(1 + r)¹⁶ = $11,000 / $9,000 = 1.2222

¹⁶√(1 + r) = ¹⁶√1.2222

1 + r = 1.01262

r = 0.01262 = 1.26% ⇒ quarterly interest rate

annual interest rate = 1.26% x 4 = 5.048%

b. Continuously

future value = present value x eᵃⁿ

• future value = $11,000
• present value = $9,000
• e = 2.718
• a = interest rate ???
• n = 4 years

$11,000 = $9,000 x 2.718⁴ⁿ

2.718⁴ⁿ = $9,000 / $11,000 = 0.818181818

⁴√2.718⁴ⁿ  =  ⁴√0.818181818

2.718ⁿ = 0.95107

nlog2.718 = log0.95107

n 0.434249452 = -0.021787543

n = -0.021787543 / 0.434249452 = -0.05017, since n must be positive, then

n = 0.05017 = 5.017%

D) a. 5.048%    b. 5.017%