A company pays a monthly rent of $3,000 per month for the office space and it pays $1,000 to each employee every month (there is a total of 12 employees). In addition each employee gets a bonus of $10 for each unit they complete during the month. The company's revenue is $18 for each unit completed by the employees. Answer the following:
1. Write the equation for the monthly salary S(x) for each employee, where x are the units completed by each employee during the month. Also, write the equation for the monthly salary for all the employees T(x).
2. If all employees completed the same number of units (x) during the month, write the equation for the company's monthly revenue R(x) as a function of x. Also, write the equation for the company's profit, P(x).
3. Supposed that one employee completed 300 units in a month, calculate the monthly salary for this employee. Additionally, suppose that each employee completed 300 units in a month, calculate the company's monthly revenue.
4. Based on the company's growth, they decided to change their location and rent a bigger office that will cost $10,000 a month. They also decided to hire 50 employees in addition to the original 12 employees. The fixed salary for the new employees and the bonus will be the same as for the original employees. The company's revenue remains the same. What will be the equation for the company's profit P(x) taking into consideration the new office and the new number of employees?
5. How many units (x) have to be completed by the employees in a month in order for the company's monthly revenue double the salary of all the employees?
1. Monthly salary of an employee = Fixed monthly salary + Bonus for units completed during the month.
S(x) = 1000 + 10*x
where x denotes the units completed by the employee during the month.
Monthly salary of all the employees = Fixed monthly salary for all the employees + Bonus for units completed during the month by all the employees.
T(x) = 12*1000 + 10*x = 10*x + 12000
where x denotes the total number of units completed by all the employees during the month.
In case, x denotes the number of units completed during the month by each employee (all employees complete same number of units), then
T(x) = 12*1000 + 10*(12*x) = 120*x + 12000
2. Company's monthly revenue = revenue per unit * number of units completed by all the employees.
R(x) = 18*(12*x) = 216*x
where x denotes the number of units completed during the month by an employee and all employees complete the same number of units during the month.
Since nothing is mentioned clearly in q5, solution works with the initial version of company with 12 employees and $3000 monthly office rent in this question. Moreover, this question is interpreted as that it is asking to double the fixed component of the monthly salary of the employees, with no change in the profit to the company.
Questions are answered in stepwise manner with sufficient explanations, and two different ways of solving q5 are explained in the solution.