Problem 77 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false. When simple interest is used, the accumulated amount is a linear function of \(t\).

Step-by-Step Solution

Answer

The statement is true. When using simple interest, the accumulated amount is a linear function of time (\(t\)), as demonstrated by the equation \(A = P + Prt\), which follows the linear form \(A(t) = mt + b\), with \(m = Pr\) and \(b = P\).

Step 1: Recall the simple interest formula

Recall the formula for calculating the accumulated amount (future value) when dealing with simple interest: \(A = P(1 + rt)\), where: \(A\) represents the accumulated amount after some time, \(P\) is the initial (principal) amount, \(r\) is the interest rate in decimal form (interest rate per annum divided by 100), and \(t\) is the time in years.

Step 2: Rewrite the formula

We can rewrite the formula, \(A = P(1 + rt)\), as \(A = P + Prt\).

Step 3: Check if the formula is a linear function

To determine if the formula we got in step 2, \(A = P + Prt\), is a linear function of \(t\), we should check if it follows the form of a linear function, which is \(f(x) = mx + b\). In this case, \(x\) corresponds to \(t\), and so the linear form of our equation should be: \(A(t) = mt + b\), where: \(m\) is the slope or rate of change, and \(b\) is the \(y\)-intercept. Comparing this linear form with the formula from step 2, we can see that the equation of the accumulated amount is indeed a linear function of \(t\). In this case, \(m = Pr\), the slope is the product of the principal and the interest rate, and \(b = P\), the initial principal amount.

Step 4: Final Conclusion

The statement is true: when simple interest is used, the accumulated amount is a linear function of time (\(t\)). This is evident from the equation \(A = P + Prt\).