mr. valdez puts $10,000 in a retirement account and does not make any deposits or withdrawals. the table…

mr. valdez puts $10,000 in a retirement account and does not make any deposits or withdrawals. the table below shows the amount of money in the account after x years. what values, rounded to the nearest hundredth, complete the exponential regression equation that models the data? years amount ($) 0 10,000 10 16,000 20 27,000 30 43,000 40 70,000 f(x) = ( )^x

mr. valdez puts $10,000 in a retirement account and does not make any deposits or withdrawals. the table below shows the amount of money in the account after x years. what values, rounded to the nearest hundredth, complete the exponential regression equation that models the data? years amount ($) 0 10,000 10 16,000 20 27,000 30 43,000 40 70,000 f(x) = ( )^x

Answer

Explanation:

Step1: Recall exponential - regression form

The general form of an exponential function is $f(x)=a(b)^x$. When $x = 0$, $f(0)=a(b)^0=a$. From the table, when $x = 0$, $f(0)=10000$, so $a = 10000$.

Step2: Use another data - point to find $b$

We use the point $(x = 10,f(10)=16000)$. Substitute $a = 10000$, $x = 10$, and $f(10)=16000$ into $f(x)=a(b)^x$. We get $16000 = 10000(b)^{10}$. First, divide both sides by 10000: $\frac{16000}{10000}=b^{10}$, so $1.6=b^{10}$. Then, solve for $b$ by taking the 10 - th root of both sides: $b = 1.6^{\frac{1}{10}}$. Using a calculator, $b\approx1.0481$.

Answer:

$f(x)=10000(1.05)^x$