Question

Find the amount owed at the end of 8 years if $5000 is loaned at a rate of 5% compounded monthly.

The half-life of silicon-32 is 710 years. If 30 grams is present now, how much will be present in 200years (Round your answer to three decimal places.)

Answer 1

1. The amount owed at the end of 8 years will be approximately $8491.72.

2. Rounding to three decimal places, there will be approximately 22.446 grams of silicon-32 present in 200 years.

1) To find the amount owed at the end of 8 years when $5000 is loaned at a rate of 5% compounded monthly, you can use the formula for compound interest:

`\[A = P(1 + (r)/(n))^(nt)\]`

Where:

• A is the final amount.
• P is the principal amount (the initial amount loaned) = $5000.
• r is the annual interest rate (in decimal form) = 5% = 0.05.
• n is the number of times the interest is compounded per year = 12 (monthly compounding).
• t is the number of years = 8.

Plug these values into the formula:

`\[A = 5000 \left(1 + (0.05)/(12)\right)^(12 \cdot 8)\]`

Now, calculate this expression:

`\[A = 5000 \left(1 + (0.05)/(12)\right)^(96)\]`

`\[A = 5000 \left(1 + (0.00416666667)/(1)\right)^(96)\]`

`\[A = 5000 \left(1.00416666667\right)^(96)\]`

Now, calculate the final amount A:

`\[A \approx 5000 * 1.69834459117\]`

`\[A \approx 8491.72\]`

So, The answer is approximately $8491.72.

2) To find how much silicon-32 will be present in 200 years if its half-life is 710 years and there are currently 30 grams, you can use the formula for exponential decay:

`\[N(t) = N_0 \left((1)/(2)\right)^{(t)/(T)}\]`

Where:

• N(t) is the amount of substance at time t.
• N0 is the initial amount = 30 grams.
• t is the time in years = 200 years.
• T is the half-life = 710 years.

Plug these values into the formula:

`\[N(200) = 30 \left((1)/(2)\right)^{(200)/(710)}\]`

Now, calculate this expression:

`\[N(200) = 30 \left((1)/(2)\right)^{(200)/(710)}\]`

`\[N(200) = 30 \left((1)/(2)\right)^(0.28169014084)\]`

`\[N(200) = 30 * 0.74820710994\]`

`\[N(200) \approx 22.4462132992\]`

The answer is approximately 22.446 grams.

Answer 2

1. The amount owed at the end of 8years is $7450

2. The amount present in 200years in three decimal places is 24.680g

1. To calculate the amount owed after 8 years , we use the formula

A =

`p(1 + (r)/(n)) ^( nt)`

Where p is the principal and r is the rate and n is the number of times compounded.

A =

`5000(1 + (0.05)/(12)) ^( 8 * 12 )`

= 5000( 1+ 0.00417)⁹⁶

= 5000( 1.00417)⁹⁶

= 5000 × 1.49

= $7450

Therefore, the amount owed at the end of 8years is $7450

2. Using the formula

N(t) = Noe⁻ˣⁿ

where x is the decay constant and no go years(time).

Decay constant = 0.693/710

= 0.000976

N(t) =

`30e { - }^(0.000976 * 200)`

N(t) = 24.680g

Therefore, there will be 24.680g after 200years