Question

brianna bought a home in 2010 for $233,250 and sold it in 2020 for $306,500. assume that the homes value increased according to an exponential model during this period.
a. what is the 10 - year growth factor for her homes value?
1.3149 preview
b. what is the 10 - year percent change for her homes value?
31.49 % preview
c. what is the 1 - year growth factor for her homes value?
1.027 preview
d. what is the 1 - year percent change for her homes value?
2.7 % preview
e. define a function f that models her homes value (in dollars) t years since 2010. make sure to use proper function notation, and do not round your growth factor too much or your answer might get marked incorrect.
f(t)=233250(1.027)^t preview f(t)=233250(1.027)^t syntax ok
submit
question 12. points possible: 3
unlimited attempts.
score on last attempt: 2.4. score in gradebook: 2.4

Explanation:

Step1: Calculate 10 - year growth factor

The 10 - year growth factor is found by dividing the final value by the initial value. Let the initial value $V_0 = 223250$ and the final value $V_{10}=306500$. The 10 - year growth factor $b_{10}=\frac{V_{10}}{V_0}=\frac{306500}{223250}=1.373$.

Step2: Calculate 10 - year percent change

The percent change formula is $\text{Percent Change}=(b_{10}- 1)\times100\%$. Substituting $b_{10}=1.373$, we get $(1.373 - 1)\times100\% = 37.3\%$.

Step3: Calculate 1 - year growth factor

If the 10 - year growth factor is $b_{10}$, and we assume an exponential model $y = a\cdot b^t$, where $t$ is in years. Let $b$ be the 1 - year growth factor. Then $b_{10}=b^{10}$. So $b = b_{10}^{\frac{1}{10}}$. Since $b_{10}=1.373$, $b = 1.373^{\frac{1}{10}}\approx1.032$.

Step4: Calculate 1 - year percent change

The 1 - year percent change is $(b - 1)\times100\%$. Substituting $b\approx1.032$, we get $(1.032 - 1)\times100\%=3.2\%$.

Step5: Define the function

The general form of an exponential function is $f(t)=a\cdot b^t$, where $a$ is the initial value and $b$ is the 1 - year growth factor. Here, $a = 223250$ and $b\approx1.032$, so $f(t)=223250\cdot(1.032)^t$.

Answer:

a. $1.373$
b. $37.3\%$
c. $1.032$
d. $3.2\%$
e. $f(t)=223250\cdot(1.032)^t$