Let's determine how long it will take the frozen yogurt machine to dispense the total amount of frozen yogurt used for each lunch period.
We know the following details:
- The machine needs [tex]\(\frac{1}{4}\)[/tex] hour (or 15 minutes) to dispense [tex]\(\frac{3}{4}\)[/tex] of the total amount of yogurt.
First, we need to find the rate at which the machine dispenses the yogurt.
1. Dispensing Rate Calculation:
The dispensing rate can be found by dividing the portion of yogurt dispensed by the time taken.
[tex]\[
\text{Dispensing Rate} = \frac{\frac{3}{4} \text{ (fraction of yogurt)}}{\frac{1}{4} \text{ (hours)}} = \frac{3/4}{1/4} = 3 \text{ (total yogurt per hour)}
\][/tex]
2. Time to Dispense Total Yogurt:
The total amount of yogurt is equivalent to 1 (since it represents the whole amount of yogurt for a lunch period). To find out the total time required to dispense all the yogurt, we use the previously calculated dispensing rate.
[tex]\[
\text{Time to dispense total yogurt} = \frac{\text{Total yogurt}}{\text{Dispensing rate}} = \frac{1 \text{ (total yogurt)}}{3 \text{ (total yogurt per hour)}} = \frac{1}{3} \text{ hour}
\][/tex]
Thus, it will take the frozen yogurt machine [tex]\(\frac{1}{3}\)[/tex] hour (or 20 minutes) to dispense the total amount of frozen yogurt used for each lunch period.
