The value
maximizes angle APB.
In this question we need to determine the <em>maximum possible</em> angle APB, which can be determined by definition of dot product, that is to say:
(1)
Where:
,
- Magnitudes of
and
.
- Internal angle, in sexagesimal degrees.
The magnitudes of each are respectively defined by line segment length formula:
,
, ![P(x, y) = (x, 0)](https://tex.z-dn.net/?f=P%28x%2C%20y%29%20%3D%20%28x%2C%200%29)
![\overrightarrow{PA} = \sqrt{(3-x)^{2}+1^{2}}](https://tex.z-dn.net/?f=%5Coverrightarrow%7BPA%7D%20%3D%20%5Csqrt%7B%283-x%29%5E%7B2%7D%2B1%5E%7B2%7D%7D)
(2)
![\overrightarrow{PB} = \sqrt{(5-x)^{2}+3^{2}}](https://tex.z-dn.net/?f=%5Coverrightarrow%7BPB%7D%20%3D%20%5Csqrt%7B%285-x%29%5E%7B2%7D%2B3%5E%7B2%7D%7D)
(3)
By (1), (2) and (3) we have the following expression:
![(3-x)\cdot (5-x) +3 = \sqrt{10-6\cdot x + x^{2}}\cdot \sqrt{34-10\cdot x + x^{2}}](https://tex.z-dn.net/?f=%283-x%29%5Ccdot%20%285-x%29%20%2B3%20%3D%20%5Csqrt%7B10-6%5Ccdot%20x%20%2B%20x%5E%7B2%7D%7D%5Ccdot%20%5Csqrt%7B34-10%5Ccdot%20x%20%2B%20x%5E%7B2%7D%7D)
![15-8\cdot x +x^{2} = \sqrt{(10-6\cdot x +x^{2})\cdot (34-10\cdot x + x^{2})}\cdot \cos \theta](https://tex.z-dn.net/?f=15-8%5Ccdot%20x%20%2Bx%5E%7B2%7D%20%3D%20%5Csqrt%7B%2810-6%5Ccdot%20x%20%2Bx%5E%7B2%7D%29%5Ccdot%20%2834-10%5Ccdot%20x%20%2B%20x%5E%7B2%7D%29%7D%5Ccdot%20%5Ccos%20%5Ctheta)
(4)
From geometry we know that sum of internal angles in triangles equals 180°, which means that angle APB must meet this condition:
![0 < \angle APB < 180](https://tex.z-dn.net/?f=0%20%3C%20%5Cangle%20APB%20%3C%20180)
In addition, we know that <em>cosine</em> function is a bounded function between -1 and 1, where
,
, ![\theta = 180^{\circ}\to -1](https://tex.z-dn.net/?f=%5Ctheta%20%3D%20180%5E%7B%5Ccirc%7D%5Cto%20-1)
A quick approach consists in graphing (4) against x. Outcome is described in the second image attached. By direct inspection, we see that
maximizes angle APB.
We kindly invite to check this question on optimization: brainly.com/question/4302495
Answer:
What subject of math is it
Step-by-step explanation:
^
Let l and w be the length and width of the rectangle. We are given the proportion 2l=4w, which implies l=2w.
The perimeter of a rectangle is given by 2(l+w). We know that the perimeter is 80, and we can plug l=2w to get
![2(l+w)=80 \iff 2(2w+w)=80 \iff 3w=40 \iff w=\dfrac{40}{3}](https://tex.z-dn.net/?f=2%28l%2Bw%29%3D80%20%5Ciff%202%282w%2Bw%29%3D80%20%5Ciff%203w%3D40%20%5Ciff%20w%3D%5Cdfrac%7B40%7D%7B3%7D)
Which, in turn, implies that
![l=2w=\dfrac{80}{3}](https://tex.z-dn.net/?f=l%3D2w%3D%5Cdfrac%7B80%7D%7B3%7D)