When one rents an apartment, one is often required to give the landlord a security deposit that is returned if the apartment is undamaged when you leave. In some localities, the landlord is required to pay the tenant interest once a year compounded yearly. Assume that the landlord is required to pay the tenant interest of 5 % compounded annually.

(a) Suppose the landlord invests a security deposit of $ 700 at a rate of 4.5 % compounded continuously. What is his net gain or loss after one year, rounded to the nearest cent? Use a negative number to denote a loss.

b) Suppose the landlord invests a security deposit of $ 700 at a rate of 6.3 % compounded continuously. What is his net gain or loss after one year, rounded to the nearest cent? Use a negative number to denote a loss.

If the landlord is investing the security deposit at a rate of 4.5% compounded continuously, then you would calculate the profits he makes using the formula P(t)=Poe^(r*t)

So,
P(t)=700e^(0.045*1)
P(t)=$732.22
The landlord makes $32.22 as a profit, but since he has to pay the tenant interest on their security deposit, you must subtract the interest as well.
P(t)=700(0.05)
P(t)=$35
Thus, if you subtract 35 from 32.22, you get $-2.78, which is the total loss.

Same thing for part B, just with different interest on the investment.

Hey there! Let's calculate the landlord's net gain or loss in these situations.

(a) If the security deposit of $700 is invested at a rate of 4.5% compounded continuously, the net gain after one year can be calculated using the formula:

A = P * e^(rt)

Where:
A = Final amount or balance
P = Principal amount (initial deposit)
e = Euler's number (approximately 2.71828)
r = Annual interest rate (in decimal form)
t = Time in years

Using the given values:
P = $700
r = 4.5% or 0.045 (since it's compounded continuously)
t = 1 year

A = 700 * e^(0.045 * 1)

After evaluating this, the landlord's net gain after one year would be approximately $736.94.

(b) Now, let's calculate the net gain or loss for a security deposit of $700 invested at a rate of 6.3% compounded continuously.

Using the same formula, but with the new interest rate:
P = $700
r = 6.3% or 0.063
t = 1 year

A = 700 * e^(0.063 * 1)

After evaluating this, the landlord's net gain after one year would be approximately $748.95.

Remember, these calculations are just for the interest gained from investing the security deposit, and they don't take into account any other factors or costs involved. Happy investing!

To calculate the net gain or loss for both scenarios, we need to use the formula for continuous compound interest:

\[A = P \cdot e^{rt}\]

Where:
A = the final amount (including interest)
P = the principal amount (the initial deposit)
e = Euler's number (approximately equal to 2.71828)
r = the interest rate (expressed as a decimal)
t = time in years

(a) For the first scenario, the principal amount (P) is $700, the interest rate (r) is 4.5% (or 0.045 as a decimal), and the time (t) is 1 year.

Plugging the values into the formula, we have:

\[A = 700 \cdot e^{0.045 \cdot 1}\]

Using a calculator, we find:

\[A \approx 741.90\]

To calculate the net gain or loss, we subtract the initial principal amount from the final amount:

Net gain or loss = A - P
Net gain or loss = 741.90 - 700
Net gain or loss ≈ 41.90

Therefore, the landlord's net gain after one year, rounded to the nearest cent, is $41.90.

(b) For the second scenario, the principal amount (P) is still $700, the interest rate (r) is 6.3% (or 0.063 as a decimal), and the time (t) is 1 year.

Plugging the values into the formula, we have:

\[A = 700 \cdot e^{0.063 \cdot 1}\]

Using a calculator, we find:

\[A \approx 746.75\]

To calculate the net gain or loss, we subtract the initial principal amount from the final amount:

Net gain or loss = A - P
Net gain or loss = 746.75 - 700
Net gain or loss ≈ 46.75

Therefore, the landlord's net gain after one year, rounded to the nearest cent, is $46.75.

To calculate the net gain or loss for the landlord, we need to compare the amount gained from the investment to the interest owed to the tenant.

(a) To find the net gain or loss when the landlord invests $700 at a rate of 4.5% compounded continuously:

Step 1: Calculate the amount gained from the investment:
Use the formula A = P * e^(rt), where A is the final amount, P is the principal amount, e is the mathematical constant approximately equal to 2.71828, r is the interest rate, and t is the time in years.

A = $700 * e^(0.045*1)
A ≈ $734.76

Step 2: Calculate the interest owed to the tenant:
The interest owed is 5% of the security deposit, which is $700 * 0.05 = $35.

Step 3: Calculate the net gain or loss:
Net gain or loss = Amount gained from the investment - Interest owed to the tenant
Net gain or loss = $734.76 - $35
Net gain or loss ≈ $699.76 (rounded to the nearest cent)

Therefore, the landlord has a net gain of approximately $699.76.

(b) To find the net gain or loss when the landlord invests $700 at a rate of 6.3% compounded continuously:

Step 1: Calculate the amount gained from the investment:
A = $700 * e^(0.063*1)
A ≈ $743.74

Step 2: Calculate the interest owed to the tenant:
The interest owed is still 5% of the security deposit, which is $700 * 0.05 = $35.

Step 3: Calculate the net gain or loss:
Net gain or loss = Amount gained from the investment - Interest owed to the tenant
Net gain or loss = $743.74 - $35
Net gain or loss ≈ $708.74 (rounded to the nearest cent)

Therefore, the landlord has a net gain of approximately $708.74.

continuous compounding ... final amount = initial amount * e^(rate * time)

(a) amount = 700 * e^(.045 * 1) ≈ 732.22