Question

The median value of a home in a particular market is decreasing exponentially. If the value of a home was initially $240,000, then its value two years later is $235,000. Answer the following.

1) Determine when the value of the home will be 90% of its original value.
Would the equation be set up like so: V=240e^.09t? 

2)Determine the rate at which the value of the home is decreasing one year after it is valued at $235,000. Include units in your answer, and round the final value to the nearest dollar.

What I know so far: V=240e^(-0.01052)t
Would you make t=1 and v=235?

3)The relative rate of change in a quantity is defined as the rate of change for that quantity divided by the quantity present. Find the relative rate of change in the home’s value at any time t.

I know that it will be dV/dt=VK
I know k=-0.01052, would the final answer be -0.-1052=(dV/dt)/V

Answer 1

1st let's calculate the decreasing rate & let V₁ be the initial value & V₂ the final's

we know that V₂=V₁.e^(r,t) where r=rate & t-time (& e=2.718)

After t= 2 years we can write the following formula

2350,000=240,000.e^(2r)==> 235,000/240000 = e^(2r) =>47/48=e^(2r)
ln(47/48)=2rlne==> ln(47/48)=2rlne=2r (since lne =1)
r= ln(47/48)/2==>r=-0.0210534/2 =-0.01052 ==> (r=-0.01052)

1) Determine when the value of the home will be 90% of its original value.
90% of 240000 =216,000
Now let's apply the formula
216,000=240,000,e^(-0.01052t), the unknown is t. Solving it by logarithm it will give t=10 years
1.a) Would the equation be set up like so: V=240e^.09t? NON, in any case if you solve it will find t=1 year

2)Determine the rate at which the value of the home is decreasing one year after : Already calculated above :(r=-0.01052)

3)The relative rate of change : it's r = -0.01052