use the following steps to find nonnegative numbers x and y that satisfy the following requirements: x + y =…

use the following steps to find nonnegative numbers x and y that satisfy the following requirements: x + y = 110 and the product p = xy is as large as possible. a) solve x + y = 110 for y. y = b) substitute the result from a) into p = xy, the equation for the variable that is to be maximized. p = c) find the domain of the function p from part b), keeping in mind the given requirements. domain is ,. d) find the derivative dp/dx, and solve the equation dp/dx = 0. dp/dx = which is equal to 0 when x = e) evaluate p at any solutions found in part d) as well as at the endpoints of the domain found in part c). at the solution found in d), the value of p is . at the left endpoint, the value of p is . at the right endpoint, the value of p is . f) state the conclusion: the maximum value of p is , attained when x = and y = .

use the following steps to find nonnegative numbers x and y that satisfy the following requirements: x + y = 110 and the product p = xy is as large as possible. a) solve x + y = 110 for y. y = b) substitute the result from a) into p = xy, the equation for the variable that is to be maximized. p = c) find the domain of the function p from part b), keeping in mind the given requirements. domain is ,. d) find the derivative dp/dx, and solve the equation dp/dx = 0. dp/dx = which is equal to 0 when x = e) evaluate p at any solutions found in part d) as well as at the endpoints of the domain found in part c). at the solution found in d), the value of p is . at the left endpoint, the value of p is . at the right endpoint, the value of p is . f) state the conclusion: the maximum value of p is , attained when x = and y = .

Answer

Explanation:

Step1: Solve for y

Given $x + y=110$, we can isolate $y$ by subtracting $x$ from both sides. $y = 110 - x$

Step2: Substitute y into P

Substitute $y = 110 - x$ into $P=xy$. $P=x(110 - x)=110x - x^{2}$

Step3: Find the domain

Since $x\geq0$ and $y = 110 - x\geq0$ (non - negative numbers), solving $110 - x\geq0$ gives $x\leq110$. So the domain is $[0,110]$.

Step4: Find the derivative and its zero

Differentiate $P = 110x - x^{2}$ with respect to $x$. Using the power rule $\frac{d}{dx}(ax^{n})=nax^{n - 1}$, we have $\frac{dP}{dx}=110-2x$. Set $\frac{dP}{dx}=0$, then $110 - 2x=0$. Solving for $x$ gives $x = 55$.

Step5: Evaluate P

  • At $x = 55$, $P=110\times55-55^{2}=55\times(110 - 55)=55\times55 = 3025$.
  • At the left - endpoint $x = 0$, $P=0\times(110 - 0)=0$.
  • At the right - endpoint $x = 110$, $P=110\times(110 - 110)=0$.

Step6: State the conclusion

The maximum value of $P$ is $3025$, attained when $x = 55$ and $y=110 - 55 = 55$.

Answer:

a) $110 - x$ b) $110x - x^{2}$ c) $0$, $110$ d) $110 - 2x$, $55$ e) $3025$, $0$, $0$ f) $3025$, $55$, $55$