health club currently charges its 1900 clients monthly membership dues of $ 42. The board of directors decides to increase the monthly membership dues. Market research shows that each $ 1 increase in dues will result in the loss of 5 clients. How much should the club charge each month to optimize the revenue from monthly​ dues?

Answer:The club should charge $211 to optimize the revenue.Step-by-step explanation:Hi there!The current revenue of the health club is calculated by multiplying the number of clients by  the membership due value.If y = revenue, then:y = 1900 · 42So, if the memberiship dues increase by $1, 5 clients will be lost. Then, if the dues increase by a value of x, then the number of lost clients will be 5x. If such an increase is applied, the revenue will be:y = (1900 - 5x) (42 + x)Apply distributive property:y = 79800 + 1900x - 210x - 5x²y = -5x² + 1690x + 79800The function is a negative parabola, then, its maximum will be the vertex of the parabola that can be calculated using the following formula:x = -b/2aWhere:b = 1690a = -5Then:x = -1690/(2 · -5)x = 169The club should charge 169 + 42 = $211 to optimize the revenue.