Problems on Antidifferentiation

[franz32]

Problem: An art collector purchased for $1000 a painting by an artist whose works are currently increasing in value with respect to time according to the formula dV/dt = 5t^(3/2) + 10t + 50, where V dollars is the anticipated value of a painting t years after its purchase. If this formula were valid for the next 6 years, what would be the anticipated value of the painting 4 years from now?

My approach: The first I did is to antidifferentiate the dV/dt. I got
V = 2t^(5/2) + 5t^2 + 50t + C. The next step.. I don't quite get it what to replace the value for t. When am I going to use 1000 for V, 6 for t and 4 for t?

I believe my answer must be higher than $1000, right? I don't how to substitute the appropriate values. Can you help me with it?

[cookiemonster]

Yes, you need to take the antiderivative.

Now apply the condition that V(0) = 1000 to determine C. This means you'll have an expression for V purely as a function of t.

Now evaluate V(4) using this expression.

cookiemonster

[jamesrc]

You would plug in t = 4 to find V(t=4), the value of the painting at time = 4 years from now. The fact that the formula is valid for 6 years is simply that, a fact that tells you that you can use you're equation to find the expected value of the painting. To find the constant C in your equation, you should assume that the collector bought the painting at market value. This means that V(t=0) = $1000. (t=0 means right... now, which reminds me of a joke, but that's neither here nor there). I hope that helps.

EDIT: sorry about that cookiemonster; I should have refreshed the page before submitting my post.

[franz32]

I got the answer, thanks.