Your grandfather purchased a house for $55,000 in 1952 and it has increased in v
ID: 2912725 • Letter: Y
Question
Your grandfather purchased a house for $55,000 in 1952 and it has increased in value according to a function y = v(x), where x is the number of years owned. These questions probe the future value of the house under various mathematical models. (Let x = 0 represent the year 1952.)
(b) Suppose the value of the house is $75,000 in 1962 and $160,000 in 1967. Assume v(x) is a quadratic function. Find a formula for v(x). (Round your values to two decimal places.)
v(x) =
What is the value of the house in 1995?
$
Using this model, in what year will the house be valued at $200,000? (Round your answer to the nearest year.)
(c) Suppose the value of the house is $75,000 in 1962. Assume v(x) is a function of exponential type. Find a formula for v(x). (Round your values to four decimal places.)
v(x) =
What is the value of the house in 1995? (Round your answer to the nearest dollar.)
$
Using this model, in what year will the house be valued at $200,000? (Round your answer to the nearest year.)
Explanation / Answer
(b) Let v(x) = ax2 +bx +c
We need to find a,b and c here
We are given v(0) = 55,000
=> 55,000 = a(0)+b(0)+c => c=55,000
Hence v(x) = ax2 +bx + 55,000
Now in 1962, x = 10 because 10 years have passed since 1952
And so v(10) = 75,000 , similarly v(15) = 160,000
=> 75000 = a(10)2 + b(10) + 55000 and 1,60,000 = a(15)2 + b(15) + 55,000
=> 20,000 = 100a + 10b and 1,05,000 = 225a +15b
Eliminating b from these two equations :
20,000*15 = 100*15 a + 10*15b and 1,05,000 * 10 = 225 *10a + 15*10 b
=> 300,000 = 1500a +150b and 1,05,0000 = 2250a + 150b
=> 1,05,0000-300,000 = 2250a-1500a
=> 750,000 = 750a => a = 1000
20,000 = 100*1000 + 10b
=> -80000 = 10b
=> b = - 8000
Hence v(x) = 1000x2-8000x+55000
In 1995, x=43 and so we find v(43) to find value of house in 1995
v(43) = 1000(43)2-8000(43)+55000
=> v(43) = 1849000-344000+55000 = 1,560,000
Hence the value in 1995 is $1,560,000
Using this model we set v(x) = 200,000 to find x
200,000 = 1000x2-8000x+55000
=> 1000x2-8000x-1,45,000=0
=> x2 - 8x -145 =0
=> x = 8+-sqrt(64+4*145)/2 { Using quadratic formula}
=> x =8+- sqrt(644)/2 = 4+-sqrt(161)
=> x = -8.68, 16.69
Discard negative value
We get x=16.69
And so somewhere in year 1969, the value of house will be $200,000
(c) If v(x) is exponential function ; then v(x) = abx
Now v(0) = 55,000
=> 55000 = ab0 => 55000=a
Hence v(x) = 55000bx
Also v(10) = 75000
=> 75000 = 55000b10
=> 75/55 = b10
=> 1.36 = b10
=> (1.36)1/10 = b
=> b = 1.03
Hence v(x) = 55000(1.03)x
In 1995, x=43
Hence v(43) = 55000(1.03)43
=> v(43) = $208721.20
Now if v(x) = 200000
200,000 = 55000(1.03)x
=> 200/55 = (1.03)x
=> 3.64 = (1.03)x
Taking log both sides :
ln(3.64) = xln(1.03)
=> x = 1.29/0.03 = 43.64
And hence after 44 years the price would be $200,000
Or in year 1996
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