Find the two numbers whose differences is 8 and the sum of their squares is a minimum
Is 11^(1/2) a rational number???
Pls help
1st number --- x
2nd number -- 8-x
sum of squares = x^2 + (8-x)^2
= x^2 + 64 - 16x + x^2
= 2x^2 - 16x + 64
d(sum of squares)/dx = 4x - 16
= 0 for max/min
4x=16
x = 4
the two numbers are both 4
check:
for the answer, sum of squares = 16+16 = 32
let 1st be 3.99, the other 4.01
sum of squares = 3.99^2 + 4.01^2 = 32.002 which is > 32
true for any other combination of x and 8-x
11^(1/2)
= √11
since 11 is not a perfect square,
11^(1/2) is irrational
To find the two numbers whose difference is 8 and the sum of their squares is a minimum, you can set up a system of equations. Let's call the two numbers x and y.
The first equation is: x - y = 8
The second equation is: x^2 + y^2 = minimum
To solve this system of equations, you can use substitution method. Rearrange the first equation to express x in terms of y: x = y + 8.
Substitute this expression for x in the second equation: (y + 8)^2 + y^2 = minimum. Expand and simplify the equation: 2y^2 + 16y + 64 + y^2 = minimum. Combine like terms to get: 3y^2 + 16y + 64 = minimum.
Now, to find the minimum value, the equation needs to be in the form of a quadratic equation, which can be achieved by factoring or completing the square. In this case, let's complete the square.
Start by dividing the equation by 3 to simplify it: y^2 + (16/3)y + (64/3) = minimum.
To complete the square, take half of the coefficient of y and square it. Half of 16/3 is 8/3, and the square of 8/3 is 64/9. Add this value to both sides of the equation:
y^2 + (16/3)y + (64/9) + (64/9) = minimum + (64/9).
y^2 + (16/3)y + 64/9 = minimum + 64/9.
Now, rewrite the left side of the equation as a perfect square trinomial:
(y + (8/3))^2 = minimum + 64/9.
Since squares are always non-negative, the left side expression has its minimum value at y = -8/3.
To find the corresponding x value, substitute y = -8/3 back into the expression x = y + 8: x = (-8/3) + 8.
Therefore, the two numbers whose difference is 8 and the sum of their squares is a minimum are x = 16/3 and y = -8/3.
Regarding your second question about whether 11^(1/2) is a rational number, the square root (√) of 11 is indeed an irrational number. This means that 11^(1/2) cannot be expressed as a fraction or ratio of two integers. Its decimal representation is approximately 3.31662479.