Mathematics, 26.06.2019 11:00, 12375819

50 ! dominique is thinking about buying a house. the table below shows the projected value of two different houses for three years. part a: what type of function, linear or exponential, can be used to describe the value of each of the houses after a fixed number of years? explain your answer. part b: write one function for each house to describe the value of the house f(x), in dollars, after x years. part c: dominique wants to purchase a house that would have the greatest value in 25 years. will there be any significant difference in the value of either house after 25 years? explain your answer, and show the value of each house after 25 years.

50 ! dominique is thinking about buying a house. the table below shows the projected value of two d

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answered: jonj75

Step-by-step explanation:

We have been given a table, which represents the projected value of two different houses for three years.

Part A:

$\text{Increase in value of house 1 after one year}=294,580-286,000$

$\text{Increase in value of house 1 after one year}=8580$

$\text{Increase in value of house 1 after two years}=303,417.40-294,580$

$\text{Increase in value of house 1 after two years}=8837.4$

We can see from our given table that the value of house 1 is not increasing at a constant rate, while a linear function has a constant rate of change, therefore, an exponential function can be used to describe the value of the house 1 after a fixed number of years.

$\text{Increase in value of house 2 after one year}=295,000-286,000$

$\text{Increase in value of house 2 after one year}=9,000$

$\text{Increase in value of house 2 after two years}=304,000-295,000$

$\text{Increase in value of house 2 after two years}=9,000$

We can see from our given table that the value of house 2 is increasing at a constant rat that is $9,000 per year. Since a linear function has a constant rate of change, therefore, a linear function can be used to describe the value of the house 2 after a fixed number of years.

Part B:

Let x be the number of years after Dominique bought the house 1.

Since value of house 1 is increasing exponentially, so let us find increase percent of value of house 1.

$\text{Increase }\%=\frac{\text{Final value-Initial value}}{\text{Initial value}}\times 100$

$\text{Increase }\%=\frac{294,580-286,000}{286,000}\times 100$

$\text{Increase }\%=\frac{8580}{286,000}\times 100$

$\text{Increase }\%=0.03\times 100$

$\text{Increase }\%=3$

$\text{Increase }\%=\frac{303,417.40-294,580}{294,580}\times 100$

$\text{Increase }\%=\frac{8837.4}{294,580}\times 100$

$\text{Increase }\%=0.03\times 100$

$\text{Increase }\%=3$

Therefore, the growth rate of house 1's value is 3%.

Since we know that an exponential function is in form: y=a*b^x , where,

a = Initial value,

b = For growth b is in form (1+r), where, r is rate in decimal form.

$3\%=\frac{3}{100}=0.03$

Upon substituting our values in exponential function form we will get,

f(x)=286,000(1+0.03)^x , where, f(x) represents the value of the house 1, in dollars, after x years.

Therefore, the function f(x)=286,000(1.03)^x represents the value of house 1 after x years.

Let x be the number of years after Dominique bought the house 2.

We can see that when Dominique bought house 2 it has a value of $286,000. This means that at x equals 0 value of house will be $286,000 and it will be our y-intercept.

Since value of house 2 is increasing 9000 per year, therefore, slope of our line be 9000.

Upon substituting these values in slope-intercept form of equation (y=mx+b) we will get,