Respuesta :
Answer
Jennifer can perform 10 Routine Physicals in the morning
Solution
- The question tells us that Jennifer works 3 1/2 hours each morning and we are asked to find how many Routine physicals she can perform in the morning if 1 Routine physical takes 1/3 hours.
- In order to find the number of Routine physicals that can be performed in 3 1/2 hours, we simply need to assume that Jennifer can perform n number of Routine physicals. That is:
[tex]\begin{gathered} \text{ Total number of Routine physicals:} \\ \frac{1}{3}+\frac{1}{3}+\frac{1}{3}+\cdots=n\times(\frac{1}{3}) \\ \text{ } \\ \text{ If we take this as the total number of Routine Physicals possible within }3\frac{1}{2}\text{ hours, and r is the number } \\ \text{ of minutes left before }3\frac{1}{2}\text{ hours are completed, we have:} \\ \\ \frac{1}{3}+\frac{1}{3}+\frac{1}{3}+\cdots+r=3\frac{1}{2}=\frac{(3\times2)+1}{2}=\frac{7}{2} \\ \\ \therefore n\times(\frac{1}{3})+r=\frac{7}{2} \end{gathered}[/tex]Thus, to find the number of Routine Physicals, n, we simply make n the subject of the formula:
[tex]\begin{gathered} n=3(\frac{7}{2}-r) \\ \\ \text{This implies that the number of Routine Physicals is divisible by 3. This means that n is a multiple of 3} \\ \text{Thus, we can test values with multiples of 3.} \\ \\ \text{when n = 3:} \\ 3\times(\frac{1}{3})=1\text{hour} \\ \\ \text{when n= 6}\colon \\ 6\times(\frac{1}{3})=2\text{hours} \\ \\ \text{when n = 9:} \\ 9\times(\frac{1}{3})=3\text{hours} \\ \\ \text{when n= 12:} \\ 12\times(\frac{1}{3})=4hours \\ \\ \text{This implies that the number of Routine Physicals cannot be greater than or equal to 12. It also } \\ \text{implies that the number of Routine Physicals cannot be less than 9 since 9 physicals would take} \\ 3\text{ hours.} \\ \\ This\text{ means that the number of possible hours is between 10 or 11.} \\ \\ when\text{ n = 10:} \\ 10\times(\frac{1}{3})=3+\frac{1}{3}\text{ hours}<3\frac{1}{2}\text{ hours} \\ \\ \text{when n = 11:} \\ 11\times(\frac{1}{3})=3+\frac{2}{3}\text{ hours}>3\frac{1}{2}\text{ hours} \\ \\ \text{Thus, we can conclude that Jennifer can only do 10 Routine Physicals in the morning} \end{gathered}[/tex]Final answer
Jennifer can perform 10 Routine Physicals in the morning
