Me on joe deposits $1,500 in an account that pays 3% annual interest compounded continuously. a. how much will joe have in his account after 5 years? b. how long will it take joe to double his money? use natural logarithms and explain your answer.

ANSWERS:

Question A. $1742.751361739

Question B. 23.1 years

WORKINGS:

Interest compounded continuously is calculated with the formula:

A = Pe^(rt)

Where,

P = Principal amount

r = Interest rate

t = Time in years

Question A. How much will Joe have in his account after 5 years

Given that Joe deposits $1,500 in the account

P = 1500

Given that the account pays 3% annual interest

r = 3/100

r = 0.03

Since we want to know how much Joe will have after 5 years

t = 5

A = Pe^(rt)

A = 1500e^(0.03 * 5)

A = 1500e^(0.15)

Using definition of logarithm:
0.15 = ln(A / 1500)
0.15 = ln(A) - ln(1500)

Add ln(1500) to both sides of the equation

0.15 + ln(1500) = ln(A) - ln(1500) +  ln(1500)
0.15 + ln(1500) = ln(A)
ln(A) = 0.15 + ln(1500)

In (A) = 0.15 + 7.313220387

In (A) = 7.463220387

Solving for A

In (A) = 7.463220387

Therefore, A = e^(7.463220387)  ;     [e = 2.718281828]

A = 1742.751361739

Therefore, Joe have $1742.751361739 in his account after 5 years

Question B. How long will it take Joe to double his money?

A = Pe^(rt)

Given that Joe deposits $1,500 in the account

P = $1500

Given that the account pays 3% annual interest

r = 3/100

r = 0.03

Since we want to know how long it will take Joe to double his money

A = $1500 * 2 = $3000

Solving for t

A = Pe^(rt)

3000 = 1500e^(0.03t)

Using definition of logarithm

0.03t = In (3000/1500)
0.03t = ln(2)

Divide both sides of the equation by 0.03

0.03t/0.03 = ln(2)/0.03
t = ln(2) / 0.03

t = 0.693147181 / 0.03

t = 23.104906019

Therefore, it will take Joe 23.1 years to double his money.

something that characterises a right triangle is the fact that we may use pythagoras' theorem to find any one of its sides given that we have the length of the other two sides. here, we have the length of the hypotenuse, √(87) yd, and one other side, √(23) yd, thus we can easily find the length of the remaining side using pythagoras' theorem as defined below:

c^2 = a^2 + b^2, where c is the length of the hypotenuse, and a and b are the lengths of the other two sides.

thus, given that c = √(87) and a = √(23), we may substitute these into the formula we defined above to get:

c^2 = a^2 + b^2

(√87)^2 = (√23)^2 + b^2 (substitute c = √87 and a = √23 into the equation)

87 = 23 + b^2 (evaluate (√87)^2 = 87 and (√23)^2 = 23 - remember that the square root of a number, squared will equal that very same number)

64 = b^2 (subtract 23 from both sides)

√64 = b (take the square root of both sides)

8 = b (evaluate √64 = 8)

therefor, the length of the unknown leg in the right triangle is 8 yd (first answer).