maximize q = xy, where x and y are positive numbers such that x + 25/3 y^2 = 16. write the objective…

maximize q = xy, where x and y are positive numbers such that x + 25/3 y^2 = 16. write the objective function in terms of y. q = 16y - 25/3 y^3 (type an expression using y as the variable.) the interval of interest of the objective function is . (simplify your answer. type your answer in interval notation.)

maximize q = xy, where x and y are positive numbers such that x + 25/3 y^2 = 16. write the objective function in terms of y. q = 16y - 25/3 y^3 (type an expression using y as the variable.) the interval of interest of the objective function is . (simplify your answer. type your answer in interval notation.)

Answer

Explanation:

Step1: Analyze variable constraints

Since (x>0) and (y > 0), from (x+\frac{25}{3}y^{2}=16), we can express (x = 16-\frac{25}{3}y^{2}>0).

Step2: Solve the inequality for (y)

[ \begin{align*} 16-\frac{25}{3}y^{2}&>0\ \frac{25}{3}y^{2}&<16\ y^{2}&<\frac{48}{25}\ -\frac{4\sqrt{3}}{5}<y&<\frac{4\sqrt{3}}{5} \end{align*} ] Combined with (y>0), we get (0 < y<\frac{4\sqrt{3}}{5}).

Answer:

((0,\frac{4\sqrt{3}}{5}))