Question

It is estimated that t years from now the value V (in dollars) of an acre of land near the ghost town of Cherokee, California, will be increasing at the rate of V(_^') (t) =〖8t〗^3/√(〖0.2t〗^4+8000) dollars per year. If the land is currently worth $500 per acre, how much will it be worth in 10 years?

Answer 1

we are given

$V'(t)=\frac{8t^3}{\sqrt{(0.2t)^4+8000}}$

we can integrate it

=dt (0.2t) 4 + 8000

we can use u-subs

u= (0.2t) + 8000

$u=0.0016\times 4t^3dt$

u=0.0064ť dt

$V(t)=\int \frac{8t^3}{\sqrt{u}}\times \frac{du}{0.0064t^3}$

$V(t)=\int \frac{8}{\sqrt{u}}\times \frac{du}{0.0064}$

$=1250\times \int \frac{1}{\sqrt{u}}du$

$=1250\times \int \:u^{-\frac{1}{2}}du$

$=2500\sqrt{u}+C$

now, we can plug back u

and we get

$V(t)=2500\sqrt{(0.2t)^4+8000}+C$

we are given

at t=0, V=500

and then we can solve for C

$500=2500\sqrt{(0.2(0))^4+8000}+C$

$C=-223106.80$

now, we can plug back C

$V(t)=2500\sqrt{(0.2t)^4+8000}-223106.80$

now, we can plug t=10

$V(10)=2500\sqrt{(0.2(10))^4+8000}-223106.80$

$V(10)=723.492$.........Answer