Find the largest possible value of
sqrt[(x-20)(y-x)] + sqrt[(140-y)(20-x)] + sqrt[(x-y)(y-140)] if
-40<=x<=100 and -20<=y<=200
Lets assume that x>y
then x>-20 (as x>y minimum value of y is -20)
see the first term
sqrt[(x-20)(y-x)]
as x>-20 and x>y
the term inside the sqrt becomes negative.
Hence x!>y (not greater than y)
similarly you can prove that y is not greater than x.
Hence x=y
put x=y in the expression
only middle term is left.
sqrt[(140-y)(20-x)]
to maximise we put x=-40
sqrt[(140-(-40))(20-(-40))]
=80
hence the answer is 80 :)
sqrt[180*60] is not 80
We have the limitation that -20 <= y <= 200, so we can't say that x = y = -40. Instead, we have to use x = y = -20 to maximize it, so sqrt(160 * 40) is 80.
Oh yeah!!! that's was a typo :P
To find the largest possible value of the given expression, we need to consider the ranges for the variables x and y and analyze the behavior of the expression within those ranges.
Given:
-40 ≤ x ≤ 100
-20 ≤ y ≤ 200
Let's consider the three individual terms in the expression and find the largest possible values for each.
Term 1: √[(x-20)(y-x)]
The maximum value for this term occurs when both factors inside the square root are maximal. Hence, we need to find the maximum of (x-20) and (y-x).
Since -40 ≤ x ≤ 100, the maximum value for (x-20) occurs at x = 100.
Also, since -20 ≤ y ≤ 200, the maximum value for (y-x) occurs at x = -40.
Therefore, the maximum value for (x-20)(y-x) is (100-20)(200-(-40)) = 120 × 240 = 28,800.
Term 2: √[(140-y)(20-x)]
Similar to Term 1, we need to find the maximum of (140-y) and (20-x).
Since -20 ≤ y ≤ 200, the maximum value for (140-y) occurs at y = -20.
Also, since -40 ≤ x ≤ 100, the maximum value for (20-x) occurs at x = 100.
Therefore, the maximum value for (140-y)(20-x) is (140-(-20))(20-100) = 160 × (-80) = -12,800.
Term 3: √[(x-y)(y-140)]
This term involves the difference between x and y. We need to find the maximum value that their difference can take.
Since -40 ≤ x ≤ 100 and -20 ≤ y ≤ 200, the maximum difference between x and y occurs when x is minimal and y is maximal. Hence, the maximum value for (x-y) is (-40-200) = -240.
Similarly, the maximum value for (y-140) occurs when y = 200.
Therefore, the maximum value for (x-y)(y-140) is (-240)(200-140) = -240 × 60 = -14,400.
Now, let's calculate the sum of the three terms:
√[(x-20)(y-x)] + √[(140-y)(20-x)] + √[(x-y)(y-140)]
= √(28,800) + √(-12,800) + √(-14,400)
= 169.705 + 113.137 + 120
Therefore, the largest possible value of the given expression is approximately 402.842.
Note: The exact value may vary depending on the precision used during the calculations.