Gasoline prices in Matthew's town have increased 18% each year for the past two years.
a.) Let g be the current price of a gallon of gas in Matthew's town. Write an equation that describes the projected price per gallon, p, at a time t years from now if this trend continues.
b.) Rewrite your equation to find the effective quarterly percentage increase, to the nearest hundredth of a percent.
a) If a current price of a gallon of gas is g, the 18% increase is 18% of g, or 0.18g. Then, if there is a 18% increase each year, at the end of one year the price per gallon will be original price plus increase: g + 0.18g = 1.18g.
At the end of the second year, the price will be increased by 18% again, so it will be the price at the end of the first year plus 18% increase:
1.18g + 0.18(1.18g) = 1.18g(1 + 0.18) = `(1.18)^2g`
At the end of the third year, it will be the price at the end of the second year plus 18% increase:`(1.18)^2g + 0.18(1.18)^2g = (1.18)^2g(1+0.18) = (1.18)^3g` .
Similarly, at the end of the t years, the projected price per gallon will be
`p = (1.18)^t g`