I didn't look through the whole problem, but shouldn't it be instead of on your very first line?
This is the question:
The value, V, in dollars, of an antique solid wood dining set t years after it is purchased can be modelled by the function
Determine the rate of change of the value of the dining set after t years.
So I think that just means find the derivative of the function?
If so this is what I did:
V'(t) = (18t^2)(0.002t^2 +1)^1/2 - (5500 +6t^3)(1/2)(0.002t^2 +1)^-1/2 (0.004t) / [(0.002t^2 +1)^1/2]^2
= (18t^2)(0.002t^2 +1)^1/2 - (5500 +6t^3)(0.002t^2 +1)^-1/2 (0.002t) / (0.002t^2 +1)
= (18t^2)(0.002t^2 +1)^1/2 - (22t +0.024t^4)(0.002t^2 +1)^-1/2 / (0.002t^2 +1)
= (0.002t^2 +1)^-1/2 [(18t^2)(0.002t^2 +1) - (22t +0.024t^4)] / (0.002t^2 +1)
= (0.002t^2 +1)^-1/2 [0.036t^4 +18t^2 -22t -0.024t^4) / (0.002t^2 +1)
= (0.002t^2 +1)^-1/2 [0.06t^4 +18t^2 -22t] / (0.002t^2 +1)
= 0.06t^4 +18t^2 -22t / (0.002t^2 +1)(0.002t^2 +1)^1/2
= t(0.06t^3 +18t -22) / (0.002t^2 +1)^3/2
Where did I go wrong? The answer is supposed to be:
= (18t^2)(0.002t^2 +1)^1/2 - (5500 +6t^3)(0.002t^2 +1)^-1/2 (0.002t) / (0.002t^2 +1)
= (18t^2)(0.002t^2 +1)^1/2 - (22t +0.024t^4)(0.002t^2 +1)^-1/2 / (0.002t^2 +1)
-> There appears to be an error between these 2 steps
(5500+6t^3)(.002t)=11t+.012t^4
-> Continuing with these values I get
(.024t^4+18t^2-11t)/(.002t^2+1)^3/2
-> Multiply by 125000/125000 *given the form of the desired result, let's just distribute 125 in the numerator and keep 1000 factored out
1000(3t^4+2250t^2-1375t)/[125000(.002t^2+1)^3/2]
->take note 125000=2500^3/2
so [125000(.002t^2+1)^3/2] = [2500(.002t^2+1)]^3/2
and distribute in the denominator also factor t out of the numerator
1000t(3t^3+2250t-1375)/(5t^2+2500)^3/2
Ok so I worked it to:
=[(0.036t^4 +18t^2) - (11t +0.012t^4)] / (0.002t^2+1)^3/2
=(0.036t^4 -0.012t^4 +18t^2 -11t) / (0.002t^2+1)^3/2
=(0.024t^4 +18t^2 -11t) / (0.002t^2+1)^3/2
I'm not entirely sure why you are multiplying by 125,000/125,000. And how did you know you would have to multiply by 125,000/125,000? It just seems like a random way to simplify further. Is it really necessary to simplify this far? 125000/125000 seems like a random number to me, how do we know thats what we have to multiply by?Multiply by 125000/125000 *given the form of the desired result, let's just distribute 125 in the numerator and keep 1000 factored out
1000(3t^4+2250t^2-1375t)/[125000(.002t^2+1)^3/2]
= 125000(0.024t^4 +18t^2 -11t) / 125000(0.002t^2 +1)^3/2
= 3000t^4 +2250000t^2 -1375000t / ???
= 1000t(3t^3 +2250t -1365) / ???
So I now get how the top part worked out.. but i'm lost with the bottom. do I also multiply 125000 by (0.002t^2 +1)^3/2? I'm not sure if your aloud to do that with a weird exponent like that.. and if I did how would I work it out to the bottom like the book answer?
The book solution probably was originally arrived at by factoring the problem before taking the derivative.
If you perform the derivative here, you should arrive directly to the same solution as the book gave you. Keep in mind your solution is not wrong by any means (unless you were asked to make all coefficients into integers).
You may ask why did we choose to multiply by 50? The answer is because it is the smallest number you can multiply that would make all the coefficients into integers.