1. If one wishes to accumulate a 30,000-fund in 5 years, how much should he deposit now at 18% compounded quarterly?
2. If a bank offers a rate of 4.5% compounded semiannually, how much should you deposit to accumulate 50,000 in 15 years
3. Suppose that a particular radioactive substance has a half-life of 25 years. Starting with 100 grams of the substance, how many grams of the substance would remain after 15 years?
Q2)25836.02212
#1 solve for x in
x(1+.18/4)^(4*5) = 30000
#2 again,
x(1+.045/2)^(2*15) = 50000
#3
100*(1/2)^(15/25)
To calculate the answers to these questions, we need to use a few formulas and concepts from finance and physics. Let's break down each question and explain how to find the answers step by step.
1. If one wishes to accumulate a $30,000 fund in 5 years, at an interest rate of 18% compounded quarterly, we can use the formula for compound interest:
A = P(1 + r/n)^(nt)
Where:
A = the future value of the investment
P = the initial deposit or principal
r = the annual interest rate (as a decimal)
n = the number of times the interest is compounded per year
t = the number of years
In this case, we want to find the initial deposit.
Let's plug in the given values into the formula:
$30,000 = P(1 + 0.18/4)^(4*5)
Now, we solve for P by isolating it:
P = $30,000 / (1 + 0.18/4)^(4*5)
You can use a calculator to evaluate the expression, and the result will give you the amount that should be deposited now to accumulate a $30,000 fund in 5 years at an 18% interest rate compounded quarterly.
2. If a bank offers a rate of 4.5% compounded semiannually, and you want to accumulate $50,000 in 15 years, we can again use the formula for compound interest.
Using the same formula as before:
A = P(1 + r/n)^(nt)
Where:
A = the future value of the investment ($50,000)
P = the initial deposit or principal (we want to find this value)
r = the annual interest rate (as a decimal) (4.5%)
n = the number of times the interest is compounded per year (in this case, semiannually, which means 2 times per year)
t = the number of years (15)
We want to solve for P:
$50,000 = P(1 + 0.045/2)^(2*15)
Isolating P:
P = $50,000 / (1 + 0.045/2)^(2*15)
Evaluate the expression using a calculator to find the amount you should deposit to accumulate $50,000 in 15 years with a 4.5% interest rate compounded semiannually.
3. To determine the grams of a radioactive substance that would remain after a certain time period, we can use the concept of half-life.
The half-life of a radioactive substance is the time it takes for half of the initial amount to decay. In this case, the half-life is 25 years.
To calculate the remaining grams after 15 years starting from 100 grams, divide the elapsed time by the half-life and raise it to the power of 2 (since the question asks for twice the half-life).
Remaining grams = 100 * (1/2)^(15/25)
Simplify and evaluate the expression to find the weight of the substance that would remain after 15 years.
Note: Keep in mind that these formulas and explanations are general guidelines. Depending on the specific context and conditions, calculations might differ, and additional factors might need to be considered.