laura is going to invest to help with a down payment on a home. how much would she have to invest to have…

Explanation:

Step1: Recall the compound - interest formula

The compound - interest formula is (A = P(1 + r)^t), where (A) is the final amount, (P) is the principal (initial investment), (r) is the annual interest rate (in decimal form), and (t) is the number of years. We need to solve for (P), so we can rewrite the formula as (P=\frac{A}{(1 + r)^t}). Given (A=$39400), (r = 0.0179) (since (1.79%=0.0179)), and (t = 10).

Step2: Substitute the values into the formula

Substitute (A = 39400), (r=0.0179), and (t = 10) into (P=\frac{A}{(1 + r)^t}). First, calculate ((1 + r)^t=(1 + 0.0179)^{10}). Using the formula (a^n=e^{n\ln(a)}), we have ((1.0179)^{10}=e^{10\ln(1.0179)}). (\ln(1.0179)\approx0.01774), then (10\ln(1.0179)\approx0.1774), and (e^{0.1774}\approx1.194). Or we can calculate ((1.0179)^{10}) directly: ((1.0179)^{10}=1.0179\times1.0179\times\cdots\times1.0179) (10 times). Using a calculator, ((1.0179)^{10}\approx1.194). Then (P=\frac{39400}{(1.0179)^{10}}=\frac{39400}{1.194}).

Step3: Calculate the value of (P)

(P=\frac{39400}{1.194}\approx33000).

Answer:

(33000)