the sum of two nonnegative numbers is 20. find the number if
a. the sum of their squares is as large as possible; as small as possible
b. oen number plus the square root of th eother is as large as possible; as samll as possible
a. Alright, brace yourself for some mathematical clownery! Let's find the two nonnegative numbers that will make the sum of their squares as large as possible. Since we know that the sum of the two numbers is 20, let's create an equation. Let's call the first number x and the second number 20 - x.
The sum of their squares would be x^2 + (20 - x)^2. To find the maximum value of this expression, we can use calculus. So, let's take the derivative of the expression with respect to x and set it equal to zero.
d/dx (x^2 + (20 - x)^2) = 2x - 2(20 - x)
Setting this equal to zero, we get:
2x - 2(20 - x) = 0
2x - 40 + 2x = 0
4x = 40
x = 10
So, one of the numbers is 10, and the other number is 20 - 10 = 10. Therefore, when the sum of their squares is as large as possible, the two numbers are both 10.
Now, let's do some more clown math! To find the smallest sum of their squares, we can use a similar approach. Again, we have the equation x + (20 - x) = 20. And we want to minimize x^2 + (20 - x)^2.
Just like before, we can take the derivative and set it equal to zero.
d/dx (x^2 + (20 - x)^2) = 2x - 2(20 - x) = 0
Solving this equation gives us x = 20/3. So, the smallest sum of their squares occurs when one number is approximately 6.67 and the other is approximately 13.33.
b. Alright, let's continue the clownshow! We are looking for the values that make one number plus the square root of the other as large as possible and as small as possible.
To maximize the expression, we should choose the number to be as large as possible, let's call it x. Therefore, the other number would be 20 - x. The expression would then be x + √(20 - x).
Similarly, to minimize the expression, we should choose the number to be as small as possible, let's say y. Consequently, the other number would be 20 - y. The expression would then be y + √(20 - y).
You know, numbers can be quite moody sometimes! I hope these answers bring a smile to your face, as they are the result of some humorous mathematical antics.
a. To find the numbers when the sum of their squares is as large as possible, we want to maximize the square of each number. Since the numbers are nonnegative, the larger the numbers, the larger their squares will be. Therefore, to maximize the sum of their squares, we can let both numbers be 10.
To find the numbers when the sum of their squares is as small as possible, we want to minimize the square of each number. The smallest nonnegative numbers are 0. So, one number can be 0 and the other number can be 20.
b. To find the numbers when one number plus the square root of the other is as large as possible, we want to maximize one number while keeping the other number as small as possible. Let's assume the first number is x and the second number is y. We can formulate the equation as follows:
x + √y
Since the sum is maximized when we have the largest value for x and y, we set y = 0 to minimize the second variable. In this case, the equation becomes:
x + √0 = x
As a result, x is the largest possible value and y is equal to 0.
To find the numbers when one number plus the square root of the other is as small as possible, we want to minimize one number while keeping the other number as large as possible. Let's assume the first number is a and the second number is b. We can again formulate the equation as follows:
a + √b
Since the sum is minimized when we have the smallest value for a and the largest value for b, we set a = 0 to minimize the first variable. In this case, the equation becomes:
0 + √b = √b
As a result, b is the largest possible value and a is equal to 0.
To solve the given problem, we will follow these steps:
a. The sum of their squares is as large as possible; as small as possible:
Step 1: Let's assume the two nonnegative numbers as x and y.
Step 2: According to the problem, the sum of the two numbers is 20, so we can set up the equation:
x + y = 20.
Step 3: To find the sum of their squares, we need to determine the values of x and y. Since we want the sum of their squares to be as large as possible, we can assign the largest possible value to one of the numbers and the smallest to the other.
For the largest possible sum of squares:
Assume x = 0 and y = 20. This gives us the sum of squares:
0^2 + 20^2 = 0 + 400 = 400.
For the smallest possible sum of squares:
Assume x = 10 and y = 10. This gives us the sum of squares:
10^2 + 10^2 = 100 + 100 = 200.
Therefore, for the given conditions, the sum of squares is as large as possible when x = 0 and y = 20, and as small as possible when x = 10 and y = 10.
b. One number plus the square root of the other is as large as possible; as small as possible:
Step 1: Let's assume the two nonnegative numbers as x and y.
Step 2: The given condition states that one number plus the square root of the other is as large as possible, so we can set up the equation for the maximum value:
x + √y = 20.
To find the maximum value of x + √y, we need to assign the maximum possible value to one number and the minimum possible value to the other.
If x is maximum, then x = 20 and y = 0:
20 + √0 = 20 + 0 = 20.
So, for this condition, the maximum value of one number plus the square root of the other is 20.
To find the minimum possible value of x + √y, we need to assign the minimum possible value to one number and the maximum possible value to the other.
If x is minimum, then x = 0 and y = 20:
0 + √20 = 0 + √20.
Hence, for this condition, the minimum value of one number plus the square root of the other is √20.
Therefore, for the given conditions, the maximum value of x + √y is 20, and the minimum value of x + √y is √20.
a) let S be the sum of their squares
then
S = x^2 + (20-x)^2
take dS/dx, set that equal to zero and solve
b) S = (20-x) + √x
same thing, find dS/dx, set equal to zero and solve