Question

X = 0.25(225 – p^2) Maximum revenue

Answer 1

Пошаговое объяснение:

Если имеется в виду максимальное значение функции, то после раскрытия скобок видим:

х=56.25-0.25р²

При р=0 х будет максимальным (0;56.25), т.к. данная функция - парабола, направленная вниз и в ее вершине будет максимальное значение функции

Answer 2

х = 56,25 - 0,25Р²

Чтобы найти максимальное значение функции, нужно взять производную и приравнять полученное выражение к нулю.

х' = -0,5Р

-0,5Р = 0, => Р = 0

Поставляем в исходное выражение: х = 56,25

Answer 3

To find the maximum revenue, we need to determine the value of p that maximizes the function. The revenue function can be expressed as:

R = X * p

Substituting the given expression for X into the revenue equation:

R = 0.25(225 – p^2) * p

R = 0.25p(225 – p^2)

To find the maximum revenue, we need to find the critical points of this function. The critical points occur when the derivative of the function is equal to zero. Let's differentiate the revenue function with respect to p:

dR/dp = 0.25(225 – p^2) - 0.25p(2p) = 0.25(225 – p^2) - 0.5p^2 = 0.25(225 - 1.5p^2)

Setting the derivative equal to zero and solving for p:

0.25(225 - 1.5p^2) = 0

225 - 1.5p^2 = 0

1.5p^2 = 225

p^2 = 150

p = ±√150

p ≈ ±12.25

Since revenue cannot be negative, we discard the negative value of p. Therefore, the critical point is p ≈ 12.25.

To confirm that this point is indeed a maximum, we can examine the second derivative of the revenue function. Let's differentiate the derivative dR/dp:

d^2R/dp^2 = -0.5p(225 - p^2)' - 0.25(225 - p^2)(2p)' = -0.5p(-3p) - 0.25(-2p) = 1.5p^2 + 0.5p

Substituting the critical point p ≈ 12.25 into the second derivative:

d^2R/dp^2 ≈ 1.5(12.25)^2 + 0.5(12.25) ≈ 221.56 + 6.13 ≈ 227.69

Since the second derivative is positive (227.69 > 0), the critical point p ≈ 12.25 corresponds to a maximum.

Therefore, the value of p that maximizes the revenue is approximately p ≈ 12.25.