Answer to Question #247677 in Math for zz2

Question

Liala and Vinay both have to travel to various locations for advertising their company's products. The company reimburses their expenses such as accommodation, food etc. The company also blacklists an employee whenever the employee's expenditure in a given month exceeds ₹ 12000. The accounts department fits the data of monthly expenditure to the polynomial E_l(x) and E_v(x) (in ₹ ) for Liala and Vinay respectively, where x is the number of months since they joined the company (i.e., x = 1 represents the completion of one month). The polynomial fit is known to be applicable for a period of 35 months (i.e., x≤35). If E_l(x) - 12,000 = b(x-5.3)(x-11)(x-24), ~b>0 and E_v(x) - 12,000 = a(x-2.5)^2(x-5.3)(x-22), ~a>0. If Vinay and Liala have been blacklisted together for at least N times in 35 months, then find the value of N.

Answer 1

$$E_l(x) - 12000 = b(x-5.3)(x-11)(x-24),$$

$$0\leq x\leq35$$

$$E_l(x) - 12000>0:x\in(5.3, 11)\cup(24, 35]$$

$$E_v(x) - 12000 = a(x-2.5)^2(x-5.3)(x-22),$$

$$0\leq x\leq35$$

$$E_v(x) - 12000>0:x\in[0, 2.5)\cup(2.5, 5.3)\cup(22, 35]$$

If Vinay and Liala have been blacklisted together, then

$$\begin{matrix} E_l(x) - 12000>0 \\ E_v(x) - 12000>0 \end{matrix}$$

 

$$x\in(24, 35]$$

Since x is integer, hen

$$x=25,26,27,28, 29, 30, 31, 32, 33, 34,35$$

There are 11 different values for $x.$

Vinay and Liala have been blacklisted together for $N=11$ times in 35 months.