Nathematics Actvanced Higher School Cert ficate Examinations by Topics compiled by profectmaths.com, au Page 20 \( 18 \quad 16 \mathrm{c} \) Kara deposits an amount of \( \$ 300000 \) into an account which pays compound interest of \( 4 \% \) per annum, added to the account at the end of each year. Immediately after the interest is added, Kara makes a withdrawal for expenses for the coming year. The first withdrawal is \( \$ P \). Each subsequent withdrawal is \( 5 \% \) greater than the previous one. Let \( \$ A_{n} \) be the amount in the account after the \( n \)th withdrawal. (i) Show that \( A_{2}=300000(1.04)^{2}-P[(1.04)+(1.05)] \) (ii) Show that \( A_{3}=300000(1.04)^{3}-P\left[(1.04)^{2}+(1.04)(1.05)+(1.05)^{2}\right] \). (i) Show that there will be money in the account when \( \left(\frac{105}{104}\right)^{n}<1+\frac{3000}{P} \).

A perfectly competitive firm has the following information in the short run. Total cost equation is given as \( T C=3 q^{2}+20 q+75 \); and the firm is currently producing 10 units of output which is the profit maximizing output level. 1. Find the profit maximizing price at which 10 units will be sold. 2. Find the level of profit or loss the firm is making in the short run. 3. What is the firm supply curve (equation)?

Calculate the gross profit margin for an item that cost the business \( \$ 3 \) and sold for \( \$ 25 \). Gross Profit = \( \$ 22 \) Gross Profit Margin = [?]\%

Calculate the gross profit margin for an item that cost the business \( \$ 36 \) and sold for \( \$ 80 \). Gross Profit \( =\$ 44 \) Gross Profit Margin \( =[?] \% \)

Calculate the gross profit margin for an item that cost the business \( \$ 12 \) and sold for \( \$ 30 \). Gross Profit = \( \$ 18 \) Gross Profit Margin = [?]\%

Calculate the gross profit margin for an item that cost the business \( \$ 20 \) and sold for \( \$ 80 \). Gross Profit \( =\$ 60 \) Gross Profit Margin \( =[?] \% \)