Find two positive real numbers such that the sum of the first number squared and the second number is 108 and their product is a maximum. (Use symbolic notation andractions where needed. Give your answer as a comma separated list of two values.)

Question

Bytelearn · Accepted Answer

Define Variables: Let's denote the first positive real number as x and the second positive real number as y. According to the problem, we have two conditions: 1. The sum of the first number squared and the second number is 108, which gives us the equation x^2 + y = 108. 2. We want to maximize the product of these two numbers, which is xy. To solve this problem, we can express y in terms of x using the first equation and then find the maximum value of the product xy using calculus or by completing the square. First, let's express y in terms of x using the first equation: y = 108 - x^2.Express y in terms of x: Now we have the product of the two numbers as a function of x: P(x) = xy = x(108 - x^2). To find the maximum value of this product, we need to take the derivative of P(x) with respect to x and set it to zero to find the critical points. Let's calculate the derivative of P(x): P'(x) = d/dx [x(108 - x^2)] = 108 - 3x^2.Calculate derivative of P(x): Next, we set the derivative equal to zero to find the critical points: 108 - 3x^2 = 0. Solving for x gives us: 3x^2 = 108, x^2 = 36, x = ±6. Since we are looking for positive real numbers, we take x = 6.Find critical points: Now we need to verify that x = 6 gives us a maximum value for the product. We can do this by checking the second derivative of P(x) at x = 6 or by observing that the function P(x) = x(108 - x^2) is an upside-down parabola, which means the critical point we found is indeed a maximum. The second derivative of P(x) is: P''(x) = d^2/dx^2 [108 - 3x^2] = -6x. Evaluating at x = 6 gives us: P''(6) = -6(6) = -36, which is less than zero, confirming that x = 6 is a maximum.Verify maximum value: Now that we have the value of x, we can find the corresponding value of y using the equation y = 108 - x^2: y = 108 - 6^2, y = 108 - 36, y = 72.Find corresponding y value: We have found two positive real numbers, x = 6 and y = 72, that satisfy the given conditions. The sum of the first number squared and the second number is 108, and their product is a maximum.

Find two positive real numbers such that the sum of the first number squared and the second number is $108$ and their product is a maximum. $\newline$ (Use symbolic notation andractions where needed. Give your answer as a comma separated list of two values.)

Full solution

More problems from Complex conjugate theorem

Related topics

Find two positive real numbers such that the sum of the first number squared and the second number is 108108108 and their product is a maximum.\newline(Use symbolic notation andractions where needed. Give your answer as a comma separated list of two values.)

Find two positive real numbers such that the sum of the first number squared and the second number is $108$ and their product is a maximum. $\newline$ (Use symbolic notation andractions where needed. Give your answer as a comma separated list of two values.)