Question

Real Estate Appreciation

Alex purchased a house 8 years ago for $42,000. This year it was appraised at $67,500.
a). A linear equation V=mt+b, 0<= t<=15, models the value V of this house for 15 years after it was purchased. Find the slope and y-intercept, and the write the equation. Know that V(0)=42,00.
b). Write and solve an equation algebraically to determine how many years after purchase this house will be worth $74,000.

Answer 1

Well, Daisy, any idea how to proceed here?

Answer 2

i know one point would be (0,48000)
and another one would be (8, 67500)

Answer 3

You have an equation template:

\[V(t) = mt + b\]and you know that \[V(0) = 42000\] and \[V(8) = 67500\]I would suggest you find the slope first, using the usual slope formula
\[m = \frac{y_2-y_1}{x_2-x_1}\]except that here you will use \[x_1 = 0\]\[x_2 = 8\]\[y_1 = 42000\]\[y_2=67500\]

Answer 4

And I agree with your points...

Answer 5

would the slope be 3187.5?

Answer 6

Yes, good.

Answer 7

so how do i know the value of y intercept?

Answer 8

Well, you know \[V(t) = 3187.5t + b\]and you know two pairs of values for \(V(t)\) and \(t\). Pick one, plug it in to that equation, and solve for \(b\).

Actually, to do the third part of the problem, it isn't strictly necessary to find the \(y-\)intercept, but as they ask you to find it...

Answer 9

I would go with \((0,42000)\) myself, as the arithmetic will be easier!

Answer 10

so the equation would be V(t)=3187.5t+42000 ?

Answer 11

Let's try out and see if it produces our two known points!

\[t=0:\]\[V(0) = 3187.5(0)+42000 = 42000\checkmark\]\[t=8:\]\[V(8) = 3187.5(8)+42000 = 25500+42000 = 67500\checkmark\]
That must be it!

Now, can you find the value of \(t\) where \(V(t) = 74000\)?

Answer 12

i got a very big number i think i didn't do it correctly

Answer 13

Good that you are skeptical of your answer. Let's think about the problem for a minute. If the slope is \(3187.5\), that means for every 1 year (increase of 1 in \(t\)), the value goes up by \(3187.5\). After 8 years, the house was worth \(67500\), so we only need another \(74000-67500 = 6500\) in appreciation, and that's roughly \(2*3187.5\), right?

Answer 14

So our answer should be in the ballpark of \(8+2\approx 10\) if we do it correctly. Want to try it again?

Answer 15

yes one moment please

Answer 16

I'm sorry, I'm kinda confused

Answer 17

Okay, here's a hint: the \(V(t)\) value is the value of the house at time \(t\). We want that to be \(74000\), so
\[74000 = 3187.5t+42000\]
Can you solve that for \(t\)?

Answer 18

is it 10.04 ?

Answer 19

well, that certainly sounds like it is in the ballpark!

\[3187.5*10.04 = 3187.5*10 + 3187.5 *0.04 \]\[=31875+ 120+4+3.2+0.28+0.02 = 32002.7\]
and after we add the \(42000\) we get \(42000+32002.7 = 74002.7\]

10.0392 would be closer, but that's probably close enough.

Answer 20

thank you so much!

Answer 21

You're welcome.