Answer:
Step-by-step explanation:
The pattern is easy to see.
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:
The measure of blank is 8
Step-by-step explanation:
Doubling the lengths of the sides of a rectangle quadruple the area
<h3>How to determine the effect?</h3>
Let the dimension of the rectangle be L and W.
So, the area is
A = LW
When the lengths are doubled, we have:
A2 = 2L * 2W
This gives
A2 = 4LW
This means that the initial area is multiplied by 4
Hence, doubling the lengths of the sides of a rectangle quadruple the area
Read more about similar shapes at:
brainly.com/question/14285697
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