Geppetto’s wish to build Pinocchio requires 5 years of regular deposits at the beginning of every month. He decides to deposit $100 on the first month with the Pleasure Island Bank, who pays interest of 6% p.a. compounded monthly. However, Geppetto’s desire to build Pinocchio is unexpectedly hard such that at the beginning of every month Geppetto makes a deposit that is 5% less than the previous deposit. At the same time, Pleasure Island Bank decides to tax 1% of Geppetto’s balance of his account at the end of month in order for them to transform more bad boys into donkeys.
Find the balance of the account at the end of the 5th year, correct to the nearest cent.
$ lies on $y=\sqrt{x-4}$, it can be expressed as $P(x_1,\sqrt{x_1-4})$. Using $m=\frac{y_2-y_1}{x_2-x_1}$, the gradient of $OP$ is $m=\frac{\sqrt{x_1-4}}{x_1}$. Since $OP$ is the tangent to $y=\sqrt{x-4}$ at $x=x_1$, you can differentiate the curve to find the gradient of $OP$ at $x=x_1$. $y=\sqrt{x-4}\to y'=\frac{1}{2\sqrt{x-4}}=\frac{1}{2\sqrt{x_1-4}}$. These two expressions for the gradient of $OP$ are equivalent, so solving $\frac{\sqrt{x_1-4}}{x_1}=\frac{1}{2\sqrt{x_1-4}}$, you should get $x_1=8$.)
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 + \int_{\frac{1}{2}}^{3} 2- \frac{3}{t+1}dt )
