Respuesta :
Answer:
The growth rate is 1.02; the value of the house in 2010 is $185,388
Step-by-step explanation:
This is an exponential growth equation, therefore, it follow the standard form:
[tex]y=a(b)^x[/tex]
where y is the value of the house after a certain number of years,
a is the initial value of the house,
b is the growth rate, and
x is the year number.
We are going to make this easy on ourselves and call year 1985 year 0. Therefore, is year 1985 is year 0, then year 2005 is year 20, and year 2010 is year 25. We will make these the x coordinates in our coordinate pairs.
(0, 113000) and (20, 155000)
Filling into our standard form using the first coordinate pair will give us the initial value of the house at the start of our problem:
[tex]113000=a(b)^0[/tex]
Anything raised to the 0 power is equal to 1, so
113000 = a(1) and
a = 113000
Now we will use that value of a along with the second pair of coordinates and solve for b, the growth rate you're looking for:
[tex]155000=113000(b)^{20}[/tex]
Start by dividing both sides by 113000 to get a decimal:
[tex]1.371681416=b^{20}[/tex]
To solve for b, we have to undo that power of 20 by taking the 20th root of b. Because this is an equation, we have to take the 20th root of both sides:
[tex]\sqrt[20]{1.371681416}=\sqrt[20]{b^{20}}[/tex]
The 20th root and the power of 20 undo each other so all we have left on the right is a b, and taking the 20th root on your calculator of the decimal on the left gives you:
b = 1.0159 which rounds to
b = 1.02 This is our growth rate.
Now we can use this growth rate and the value of a we found to write the model for our situation:
[tex]y=113000(1.02)^x[/tex]
If we want to find the value of the house in the year 2010 (year 25 to us), we sub in a 25 for x and do the math:
[tex]y=113000(1.02)^{25}[/tex]
Raise 1.02 to the 25th power and get:
y = 113000(1.640605994) and multiply to get a final value of
y = $185,388
Answer:
(A) The growth rate is 1.02
(B) The value of house in year 2025 will be $185388.
Step-by-step explanation:
Given information:
Value of house in 1985 was $113,000
Value of house in 2005 is $155,000
As, it is an exponential growth so we use the equation;
[tex]y=ab^x[/tex]
Here, [tex]a[/tex] = initial value of the house
and, [tex]x[/tex] = years in number.
Now, according to given information,
The points can be written as (0,113000) and (20,155000) because the gap is of 20 years.
Hence, the equation will be;
[tex]11300=ab^0[/tex]
here [tex]b^0[/tex] will be 1.
So [tex]a=113000[/tex]
and
[tex]155000=ab^{20}[/tex]
on dividing both side by 113000.
[tex]1.3716=b^{20}\\b=\sqrt[20]{1.3716}\\b=1.02[/tex]
Hence , the growth rate is 1.02
Now , for value in 2025 of the house
put the values in the equation as :
[tex]b=1.02\\x=25[/tex]
[tex]y=113000(1.02)^{25}\\y=113000 \times 1.65\\y=185388[/tex]
Hence, the value of house in year 2025 will be $185388.
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