Question

in the year 1985, a house was valued at $110,000. By the year 2005, the value had appreciated to $145,000. Find the annual growth rate between 1985 and 2005 assuming that the value continued to griw by the same percentage snd use it to write a function that models the exponential growth in the house’s value in dollars V over time t, where t is the number if years since 1985.Write your answer in the form V(t)=a(b)^t. Round your b value to 4 decimal places. Do not include commas in your answer.

Answer 1

SOLUTION:

Case: Exponential Equations

Exponentiation is a mathematical operation, written as bⁿ, involving two numbers, the base b, and the exponent or power n, and pronounced as "b raised to the power of n

Given:

In 1985, the house was valued at $110,000,

V(t)=?

1985 is the starting point, t=0

V(0)= $110,000

In 2005, the house was valued at $145,000,

2005 is the starting point, t=

V(20)= $145,000

Method:

$V(t)=a(b)^t$

Step 1: Plug V(0)= $110,000

$\begin{gathered} V(0)=a(b)^0 \\ 110000=a\times1 \\ a=110000 \end{gathered}$

Step 2: Now we replace the value of a

$V(t)=110000(b)^t$

Step 3: Now we plug V(20)= $145,000

$\begin{gathered} V(20)=110000(b)^{20} \\ 145000=110,000(b)^{20} \\ Divide\text{ both sides by 110000} \\ 1.3181818=b^{20} \\ b=\sqrt[20]{1.3181818} \\ b=1.0139 \end{gathered}$

Step 4: Therefore the model has the equation:

$V(t)=110000(1.0139)^t$

Final answer:

$V(t)=110,000(1.013,9)^{t}$