Question

Difficulty on the Reihman scale of really tough problems: 2/10 (+2 pts)

Reihman and Van Ausdall both decide to open retirement accounts. Mr. Van Ausdall puts in $35,000 with a continuous rate of 5% and Mr. Reihman puts in $10,000 with a continuous rate of 7%. After how much time will Mr. Reihman's account outgrow Mr. Van Ausdall? Solve algebraically and show all work.

Answer 1

Mr. Reihman's account will outgrow Mr. Van Ausdall's account after approximately 51.54 years.

Explanation:

We can use the formula A = Pe^(rt) to calculate the balance (A) of each account, where P is the principal, e is the base of the natural logarithm (approximately 2.71828), r is the continuous interest rate, and t is the time in years.

For Mr. Van Ausdall's account, we have:

A = 35,000*e^(0.05t)

For Mr. Reihman's account, we have:

A = 10,000*e^(0.07t)

We want to find the time when Mr. Reihman's account outgrows Mr. Van Ausdall's account. This means we want to find the value of t such that:

10,000e^(0.07t) > 35,000e^(0.05t)

Dividing both sides by 10,000*e^(0.05t), we get:

e^(0.02t) > 3.5

Taking the natural logarithm of both sides, we get:

0.02t > ln(3.5)

Solving for t, we get:

t > ln(3.5)/0.02 ≈ 51.54

Therefore, Mr. Reihman's account will outgrow Mr. Van Ausdall's account after approximately 51.54 years.