Sierra King is a nail technician. She allots 20 minutes for a manicure and 45 minutes for a pedicure in her 7-hour work day. No more than 5 pedicures can be scheduled each day. The prices are $18 for a manicure and $45 for a pedicure. How many manicures and pedicures should Ms. King maximize to schedule her daily income? What is her maximum daily income?
To maximize Ms. King's daily income, we need to find the optimal number of manicures and pedicures she should schedule. Let's break down the problem and find the solution step by step:
1. Calculate the maximum number of pedicures that can be scheduled each day:
Ms. King can spend a maximum of 7 hours (420 minutes) for work each day. Each pedicure takes 45 minutes, so we can calculate the maximum number of pedicures as:
Maximum pedicures = 420 minutes / 45 minutes = 9.33 (approximately)
Since she can't schedule a fraction of a pedicure, the maximum number of pedicures is limited to 5.
2. Calculate the time available for manicures:
The remaining time after scheduling the maximum number of pedicures is:
Remaining time = 420 minutes - (45 minutes * 5 pedicures) = 195 minutes.
Each manicure takes 20 minutes, so we can calculate the maximum number of manicures as:
Maximum manicures = 195 minutes / 20 minutes = 9.75 (approximately)
Since she can't schedule a fraction of a manicure, the maximum number of manicures is limited to 9.
3. Calculate the maximum daily income:
The price for a manicure is $18, and the price for a pedicure is $45. To calculate the maximum daily income, multiply the number of manicures by the manicure price and the number of pedicures by the pedicure price, and then sum them up:
Maximum daily income = (Price of manicure * Number of manicures) + (Price of pedicure * Number of pedicures)
= ($18 * 9) + ($45 * 5)
= $162 + $225
= $387
Therefore, to maximize her daily income, Ms. King should schedule 9 manicures and 5 pedicures, resulting in a maximum daily income of $387.
To maximize Ms. King's daily income, we need to determine the number of manicures and pedicures she should schedule within her 7-hour workday.
Let's assume she schedules m manicures and p pedicures.
1. Time available for manicures can be calculated as:
Total work hours - time allotted for pedicures = 7 hours - (p × 45 minutes)
2. Time available for manicures in minutes can be converted as:
(7 - (p × 45)) × 60 minutes/hour = (420 - 45p) minutes
3. Time required for each manicure is given as 20 minutes.
4. The maximum number of manicures she can schedule would be:
(420 - 45p)/20 = (21 - (9/4)p) manicures
5. Since Ms. King can schedule a maximum of 5 pedicures per day, the range for p is 0 to 5.
Using this information, we can calculate the maximum daily income.
Daily income for manicures = (number of manicures) × (price per manicure) = [(21 - (9/4)p) × $18]
Daily income for pedicures = (number of pedicures) × (price per pedicure) = (p × $45)
Total daily income = (daily income for manicures) + (daily income for pedicures) = [(21 - (9/4)p) × $18] + (p × $45)
To find the maximum daily income, we can substitute each value of p (0 to 5) and calculate the corresponding daily income.
Let's calculate the daily income for each value of p:
For p = 0:
Daily income = [(21 - (9/4) * 0) × 18] + (0 × 45) = (21 × 18) = $378
For p = 1:
Daily income = [(21 - (9/4) * 1) × 18] + (1 × 45) = (19.75 × 18) + 45 = $388.50
For p = 2:
Daily income = [(21 - (9/4) * 2) × 18] + (2 × 45) = (18.5 × 18) + 90 = $387
For p = 3:
Daily income = [(21 - (9/4) * 3) × 18] + (3 × 45) = (17.25 × 18) + 135 = $384.75
For p = 4:
Daily income = [(21 - (9/4) * 4) × 18] + (4 × 45) = (16 × 18) + 180 = $384
For p = 5:
Daily income = [(21 - (9/4) * 5) × 18] + (5 × 45) = (14.75 × 18) + 225 = $383.50
The maximum daily income is obtained when p = 1, and the maximum number of pedicures she can schedule is 1.
Therefore, Ms. King should schedule 1 manicure and 1 pedicure to maximize her daily income, which is $388.50.