Question

At the beginning of 2000 Randall's house was worth 234 thousand dollars and Damien's house was worth 110 thousand dollars. At the beginning of 2003, Randall's house was worth 176 thousand dollars and Damien's house was worth 154 thousand dollars. Assume that the values of both houses vary at an exponential rate.
Write a function $f$ ,that determines the value of Randall's house (in thousands of dollars) in terms of the number of years $t$ since the beginning of 2000.

Write a function $g$ that determines the value of Damien's house (in thousands of dollars) in terms of the number of years $t$ since the beginning of 2000.

How many years after the beginning of 2000 will Randall's and Damien's house have the same value?

Answer 1

Using linear functions, we have that:

• Randall's value is: f(t) = -19t + 234.

• Damien's value is: g(t) = 14.67t + 110

• They will have the same value in 3.68 years since the start of 2000.
  

What is a linear function?

A linear function is modeled by:

y = mx + b

In which:

• m is the slope, which is the rate of change, that is, by how much y changes when x changes by 1.

• b is the y-intercept, which is the value of y when x = 0, and can also be interpreted as the initial value of the function.

We consider the initial values as the y-intercepts. Randall's house decayed 58 thousand dollars in 3 years, hence the slope is:

m = -58/3 = -19.

Hence the function for the value, in thousands of dollars, is:

f(t) = -19t + 234.

Damien's initial value was of 110, and it increased 44 thousand in 3 years, hence the slope is:

m = 44/3 = 14.67

Hence the function for the value of Damien's house, in thousands of dollars, in t years after 2003, is:

g(t) = 14.67t + 110

They will have the same value when:

f(t) = g(t)

-19t + 234 = 14.67t + 110

33.67t = 124

t = 124/33.67

t = 3.68

They will have the same value in 3.68 years since the start of 2000.

More can be learned about linear functions at brainly.com/question/24808124

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