Answer:
x = √(a(a+b))
Step-by-step explanation:
We can also assume a > 0 and b > 0 without loss of generality. (If a and a+b have opposite signs, the maximum angle is 180° at x=0.)
We choose to define tan(α) = -(b+a)/x and tan(β) = -a/x. Then the tangent of ∠APB is ...
tan(∠APB) = (tan(α) -tan(β))/(1 +tan(α)tan(β))
= ((-(a+b)/x) -(-a/x))/(1 +(-(a+b)/x)(-a/x))
= (-bx)/(x^2 +ab +a^2)
This will be maximized when its derivative is zero.
d(tan(∠APB))/dx = ((x^2 +ab +a^2)(-b) -(-bx)(2x))/(x^2 +ab +a^2)^2
The derivative will be zero when the numerator is zero, so we want ...
bx^2 -ab^2 -a^2b = 0
b(x^2 -(a(a+b))) = 0
This has solutions ...
b = 0
x = √(a(a+b))
The former case is the degenerate case where ∠APB is 0, and the value of x can be anything.
The latter case is the one of interest:
x = √(a(a+b)) . . . . . . the geometric mean of A and B rotated to the x-axis.
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<em>Comment on the result</em>
This result is validated by experiments using a geometry program. The location of P can be constructed in a few simple steps: Construct a semicircle through the origin and B. Find the intersection point of that semicircle with a line through A parallel to the x-axis. The distance from the origin to that intersection point is x.
Answer:
1/4 probability of getting a product that isn't divisible by 2.
Step-by-step explanation:
These are all the possible outcomes
1 x 1 = 1 2 x 1 = 2 3 x 1 = 3 4 x 1 = 4 5 x 1 = 5 6 x 1 = 6
1 x 2 = 2 2 x 2 = 4 3 x 2 = 6 4 x 2 = 8 5 x 2 = 10 6 x 2 = 12
1 x 3 = 3 2 x 3 = 6 3 x 3 = 9 4 x 3 = 12 5 x 3 = 15 6 x 3 = 18
1 x 4 = 4 2 x 4 = 8 3 x 4 = 12 4 x 4 = 16 5 x 4 = 20 6 x 4 = 24
1 x 5 = 5 2 x 5 = 10 3 x 5 = 15 4 x 5 = 20 5 x 5 = 25 6 x 5 = 30
1 x 6 = 6 2 x 6 = 12 3 x 6 = 18 4 x 6 = 24 5 x 6 = 30 6 x 6 = 36
All of the outcomes that aren't divisible by 2 are in bold
There are 9 out of 36 possible outcomes that aren't divisible by 2
9/36 = 1/4
I just need points I’m sorry :(
Answer:
-12
Step-by-step explanation:
6 ÷ -1/2
Copy dot flip
6 * -2/1
-12
