Question

pria bought a new home in 2012. the value of the home increased exponentially each year. the value of the home in 2014 was $227,136. in 2016, the home was worth $245,670. how much did pria pay for the home in 2012? (1 point) $190,068 $245,670 $227,136 $210,000 graphing calculator

Explanation:

Step1: Define the exponential growth model

The general form of an exponential growth function is \( V(t) = V_0 \cdot r^t \), where \( V(t) \) is the value at time \( t \), \( V_0 \) is the initial value (in 2012, so \( t = 0 \) corresponds to 2012), \( r \) is the growth rate, and \( t \) is the number of years since 2012.

In 2014, \( t = 2 \) (since 2014 - 2012 = 2) and \( V(2) = 227136 \). In 2016, \( t = 4 \) (2016 - 2012 = 4) and \( V(4) = 245670 \).

Step2: Use the two points to find \( r \)

We know that \( V(2) = V_0 \cdot r^2 = 227136 \) and \( V(4) = V_0 \cdot r^4 = 245670 \).

Notice that \( r^4=(r^2)^2 \), so we can divide the equation for \( V(4) \) by the equation for \( V(2) \):

\[
\frac{V_0 \cdot r^4}{V_0 \cdot r^2}=\frac{245670}{227136}
\]

Simplify the left side: \( r^2=\frac{245670}{227136} \)

Calculate \( \frac{245670}{227136} \approx 1.0816 \) (we can also simplify the fraction: divide numerator and denominator by 6, \( \frac{245670\div6}{227136\div6}=\frac{40945}{37856} \approx 1.0816 \))

So \( r^2 = 1.0816 \), then \( r=\sqrt{1.0816} \approx 1.04 \) (since \( r>0 \) for growth)

Step3: Find \( V_0 \)

Now use \( V(2) = V_0 \cdot r^2 = 227136 \), and we know \( r^2 = 1.0816 \) (or we can use \( r = 1.04 \), so \( r^2=(1.04)^2 = 1.0816 \))

So \( V_0=\frac{227136}{r^2}=\frac{227136}{1.0816} \)

Calculate \( \frac{227136}{1.0816}=210000 \) (because \( 210000\times1.0816 = 210000\times(1 + 0.0816)=210000+210000\times0.0816 = 210000 + 17136=227136 \))

We can also check with \( t = 4 \): \( V_0\cdot r^4=210000\times(1.04)^4 \). Calculate \( (1.04)^4=(1.04^2)^2=(1.0816)^2 = 1.16985856 \). Then \( 210000\times1.16985856 = 210000\times1.16985856 = 245670 \) (which matches the value in 2016). So \( V_0 = 210000 \)

Answer:

\( \$210,000 \) (the option: \( \$210,000 \))