The sum of two nonnegative numbers is 36. Find the numbers if

Question

a. the difference of their square roots is to be as large as possible.

b. the sum of their square roots is to be as large as possible.

Accepted Answer

Two tanks contain a total of 120 L of water. Find the volumes of water in each tank if the difference of the square roots is to be as large as possible. And we have 4 possible answers, being 0 and 120, 60 and 60, 40 and 80, or 55 and 65, all in liters. Now, to solve this, let's first create our volumes. Our first volume Will just be X. Our second volume Will be 120 minus x. Because the sum of both of them should equal 120. Now, it says we want to maximize the difference of the square roots. We didn't make a relationship. Where X will be less standard equals to 120 minus X, making V2 the larger volume. This will help us find our interval. If we were to solve this, we would get 2 X is less than or equals to 120 by moving the negative X to the other side. And dividing by 2, gives us X is less than equals to 60. Now, we also can't have a negative volume. So X will be on the interval. From 0 to 60. No. Let's maximize their square roots. Our function F of X will be the difference of square roots. The square root of 120 minus X. Minus the square root of x. Now, to maximize this difference, we should look for the critical point. Now, to find the critical point, we will take the derivative of our equation and set it equals to zero. So we have the derivative DDX. Square root, 120 minus X minus the square root of X is equals to 0. Now let's solve this derivative. These are both chain rule derivatives. The dreli Of the square root of X is 1 divided by 2 square roots of X. This is known. So we can apply this to our derivative. We have 1 divided by 2. Radical 120 minus X. Multiplied by the interior derivative. -1. Well this 1 divided by 2 square roots of X. This gives us -1 divided by 2 square roots of 120 minus X. Minus 1 divided by 2 squares of X. That equals 0. Now, if you multiply both sides by negative 2, we will then end up Getting 1 divided by the square root of 120 minus X, plus 1 divided by the square root of X. This is equals to zero. Now both of these functions are always positive. Which means there's no place that will allow this function to equal 0. This means there are no critical points. So when there are no critical points, we can check the boundary. Since we're on the interval from 0 to 60, We will check X equals 0 and X equals 60. We have the square root Of 120 0. Minus the square root of 0. Which simplifies To 2 square roots of 30. For x equals 60. We have F of 60 equals the square root. Of 120. Minus 60 minus the square root of 60, which just gives us 0. Now, we noticed then that the maximum value occurs when X equals 0, giving us 2 squares of 30. We can then tell that X equals 0 L. And because X equals 0, our other volume was 120 minus X. Our other volume should be 120. Leaders. OK, I hope to help you solve the problem. Thank you for watching. Goodbye.

The sum of two nonnegative numbers is 36. Find the numbers if
a. the difference of their square roots is to be as large as possible.
b. the sum of their square roots is to be as large as possible.

Key Concepts

Optimization

Constraints

Square Roots

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