Identify the point of intersection of these two curves:
P(t)=300(1.05)^t
F(t)=1000(0.92)^t
300(1.05)^t=1000(.92^t)
take the log of each side..
log 300+tlog1.05=log1000+tlog.92
t(log1.05-log.92)=log 1000-log300
t(log(1.05/.92)=log(1000/300)
t= log(920/315)=.465
check my math.
you did it right up until:
t = log(920/314) = 0.465, I have no idea how you got those numbers but it's
t(log(1.05/.92))=log(1000/300)
t(log(1.1141304348)=log(3.3(rpt.))
t(0.057401472)=0.522878745
t=(0.522878745)/(0.057401472)
t=9.1092
To find the point of intersection of the two curves P(t) and F(t), we need to set them equal to each other and solve for t.
So, we have:
300(1.05)^t = 1000(0.92)^t
To solve this equation, we can take the logarithm of both sides. Let's take the natural logarithm (ln) of both sides:
ln(300(1.05)^t) = ln(1000(0.92)^t)
Using the properties of logarithms, we can simplify this equation:
ln(300) + ln(1.05^t) = ln(1000) + ln(0.92^t)
Now, using the logarithmic identity ln(a^b) = b * ln(a), we can further simplify:
ln(300) + t * ln(1.05) = ln(1000) + t * ln(0.92)
Now, let's isolate the variable t:
t * ln(1.05) - t * ln(0.92) = ln(1000) - ln(300)
Factoring out t:
t * (ln(1.05) - ln(0.92)) = ln(1000) - ln(300)
Finally, divide both sides by (ln(1.05) - ln(0.92)) to solve for t:
t = (ln(1000) - ln(300))/(ln(1.05) - ln(0.92))
Using a calculator, we can obtain the value of t. Once we have the value of t, we can substitute it into either of the original equations (P(t) or F(t)) to find the corresponding y-value. This will give us the point of intersection of the two curves.