# Find the accumulated value at time 8 of $1600 invested at time 6 Given:$a(t) = xt^{2} + yt + z 100a(2) = 152$and$200a(4) = 240$Using that, I found that$z = 1, x = -0.105 $, and$y = 0.47$The question asks for the accumulated value at time 8 of 1600 invested at time 6. So we need to find what was invested at time 0 for it to equal 1600 at time 6:$1600= P(-0.105*6^{2} + 0.47*6 + 1) $=>$P = 40,000$This is already wrong... But I don't know why. I can't find A(8) without that P. ## 1 Answer Actually everything you've done so far is right (however there's technically no need to calculate P, since the answer would simply be$1600\times\dfrac{A(8)}{A(6)}$). With that in mind, if you analyze$A(t)$with the coefficients you've already found, i.e.$A(t)=-0.105t^2+0.47t+1$, you'll notice there is a turning point at$t=2.238$, that$A(t)$is decreasing for$t>2.238$, and that there is a zero at$t=6.05$. If you check, you'll get$A(6) = 0.04$, which matches up with what you found above (i.e. it gives$P=40,000$). Your conceptual error seems to be that you're assuming$A(t)$must always be greater than 1 (which would usually be the case, but definitely isn't in this question). So given all of the above, we can quite easily find that$A(6)=0.04$, and$A(8)=-1.96$, and therefore the answer is$1600\times\dfrac{A(8)}{A(6)}=1600\times\dfrac{-1.96}{0.04}=-\$78,400$

• The answer I was given is \$2057.14. I guess I will assume they made a typo. Also, why is the answer 1600*A(8)/A(6). How did you rearrange the formula to get that?
– Bob
Feb 4, 2015 at 17:27
• Hopefully I'm not jumping the gun on your learning here, but we can represent A(t) as A(0,t) as well, which indicates that it is the accumulation factor from time 0 to time t. Now if you think of A(2), you can also write it as A(0,1)*A(1,2) - that's how the accumulation works. So in general, A(0,t)=A(0,s)*A(s,t). We can rearrange that to get A(s,t)=A(0,t)/A(0,s), or in words, the accumulation factor from time 6 to time 8 is A(0,8)/A(0,6)=A(8)/A(6). So the 1600*A(8)/A(6) actually comes from 1600*A(6,8), which should seem much more logical. Hope that helps. Feb 4, 2015 at 22:23