Find two positive real numbers whose product is a maximum. The sum is 110.

Question

Witionsion · Accepted Answer

Let x and y be the two positive real numbers. Write an equation to show that the sum of the two positive real numbers is 110.
$x + y = 110$
Solve for the value of y.
$x + y = 110$
$x + y - x = 110 - x$
$y = 110 - x$
Write an equation to show that the product of the two real numbers is the maximum. Let f(x) be the maximum. Substitute the values of x and y. Write as a quadratic function, $f (x) = a x^{2} + b x + c$
$f (x) = x y$
$f (x) = x (110 - x)$
$f (x) = 110 x - x^{2}$
$f (x) = - x^{2} + 110 x$
$a = - 1$ which is $a < 0$ , the maximum at $x = - \frac{b}{2 a}$ . Detemine the value of x. Let a=-1 b=110
$x = - \frac{b}{2 a}$
$x = - \frac{110}{2 (- 1)}$
$x = - \frac{110}{- 2}$
$x = 55$
Substitute the value of x in $y = 110 - x$
$y = 110 - x$
$y = 110 - 55$
$y = 55$
The two positive real numbers are 55 and 55

nick1337 · Accepted Answer

Step 1:To find two positive real numbers whose product is a maximum and whose sum is 110, we can solve this problem using calculus. Let x and y be the two positive real numbers.We are given the constraint that x+y=110. We need to find the values of x and y that maximize their product P=xy.To solve this problem, we can use the method of Lagrange multipliers. We want to maximize the function P=xy subject to the constraint g(x,y)=x+y−110=0.The Lagrangian function is defined as L(x,y,λ)=xy−λ(x+y−110), where λ is the Lagrange multiplier.To find the critical points, we take the partial derivatives of L with respect to x, y, and λ, and set them equal to zero:∂L∂x=y−λ=0∂L∂y=x−λ=0∂L∂λ=x+y−110=0From the first two equations, we can solve for x and y in terms of λ:x=λy=λStep 2:Substituting these values into the third equation, we have:2λ−110=0Solving for λ, we get λ=55.Using the values of λ obtained, we can find x and y:x=λ=55y=λ=55Therefore, the two positive real numbers that maximize their product while summing up to 110 are x=55 and y=55.To summarize, the solution is x=55 and y=55, with their product being P=xy=55·55=3025.

Vasquez · Accepted Answer

Answer:x=55 and y=55Explanation:The product of x and y can be expressed as P=xy. To find the maximum product, we can use the concept of the arithmetic-geometric mean inequality. According to this inequality, the arithmetic mean is always greater than or equal to the geometric mean.The arithmetic mean of x and y is given by x+y2, and the geometric mean is given by xy. Therefore, we have the inequality:x+y2≥xyTo maximize the product P, we want the left side of the inequality to be equal to the right side. Therefore, we can rewrite the inequality as an equation:x+y2=xyTo solve this equation, let&#039;s square both sides:(x+y2)2=(xy)2Simplifying, we have:x2+2xy+y24=xyMultiplying both sides by 4, we get:x2+2xy+y2=4xyRearranging the terms, we have:x2−2xy+y2=0Factoring, we obtain:(x−y)2=0Taking the square root of both sides, we have:x−y=0Solving for y, we find:y=xSince we are given that the sum of x and y is 110, we can write the equation:x+x=110Simplifying, we get:2x=110Dividing both sides by 2, we obtain:x=1102Hence, x=55.Since y=x, we also have y=55.Therefore, the two positive real numbers whose product is a maximum and whose sum is 110 are x=55 and y=55.

RizerMix · Accepted Answer

Let x and y be the two positive real numbers. Write an equation to show that the sum of the two positive real numbers is 110.
$x + y = 110$
Solve for the value of y.
$x + y = 110$
$x + y - x = 110 - x$
$y = 110 - x$
Write an equation to show that the product of the two real numbers is the maximum. Let f(x) be the maximum. Substitute the values of x and y. Write as a quadratic function, $f (x) = a x^{2} + b x + c$
$f (x) = x y$
$f (x) = x (110 - x)$
$f (x) = 110 x - x^{2}$
$f (x) = - x^{2} + 110 x$
$a = - 1$ which is $a < 0$ , the maximum at $x = - \frac{b}{2 a}$ . Detemine the value of x. Let a=-1 b=110
$x = - \frac{b}{2 a}$
$x = - \frac{110}{2 (- 1)}$
$x = - \frac{110}{- 2}$
$x = 55$
Substitute the value of x in $y = 110 - x$
$y = 110 - x$
$y = 110 - 55$
$y = 55$
The two positive real numbers are 55 and 55

Find two positive real numbers whose product is a maximum.

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