Question

Two high school students are hired to rake leaves. They will work a couple hourseach afternoon until the job is completed. They can choose one of two paymentplans. Plan A pays $11.50 per afternoon, while Plan B pays 2 cents for one day ofwork, 4 cents total for two days of work, 8 cents total for three days of work, 16cents total for four days, and so on. On which day would their total pay be approximately the same?

Answer 1

$\begin{gathered} \text{Plan A: This plan has equation y=11.5x} \\ \text{Plan B: This plan has equation y=}\frac{\text{2}^x}{100} \\ \text{where x is the day and y is payment. For example, if they works 2 days:} \\ \text{Plan A: y=11.5(2)}\Rightarrow y=23\text{ dollars} \\ \text{Plan B: y=}\frac{\text{2}^2}{100}=\frac{2\cdot2}{100}\Rightarrow y=0.04\text{ dollars} \\ As\text{ you can note, 100 divides } \\ 2^x\text{ in order to convert cents to dollars} \end{gathered}$$\begin{gathered} \text{The total pay will be the same when y in Plan A is equal to the value of y in Plan B, that is,} \\ 11.5x=\frac{2^x}{100} \\ \text{Then, we must find x. Hence,} \\ (100)(11.5x)=2^x \\ 1150x=2^x \\ By\text{ solving numerically this equation, the answer is} \\ x=13.97 \\ By\text{ rouding up, x=14 day} \end{gathered}$