The daily profit p in dollars of a company making tables is described by the function upper p left parenthesis x right parenthesis equals negative 5 x squared plus 240 x minus 2475 p(x)=? 5 x 2+240x? 2475? , where x is the number of tables that are manufactured in 1 day. the maximum profit of the company occurs at the vertex of the parabola. how many tables should be made per day in order to obtain the maximum profit for the? company

The following equation of parabola is given:

p(x)= - 5 x^2 + 240 x - 2475

where p(x) = y

This is a standard form of the parabola. We need to convert this into vertex form of equation. The equation must be in the form:

y – k = a (x – h)^2

Where h and k are the vertex of the parabola. Therefore,

y = - 5 x^2 + 240 x - 2475

y = -5 (x^2 – 48 x + 495)

Completing the square:

y = -5 (x^2 - 48 x + 495 + _) - (-5)* _

Where the value in the blank _ is = -b/2

Since b = -48        therefore,

y = -5 (x^2 – 48 x + 495 + 81) + 405

y – 405 = -5 (x^2 – 48 x + 576)

y – 405 = -5 (x – 24)^2

Therefore the vertex is at points (24, 405).

The company should make 24 tables per day to attain maximum profit.

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