Answer:
![\angle D=37.7^{\circ}](https://tex.z-dn.net/?f=%5Cangle%20D%3D37.7%5E%7B%5Ccirc%7D)
![\angle E=52.3^{\circ}](https://tex.z-dn.net/?f=%5Cangle%20E%3D52.3%5E%7B%5Ccirc%7D)
![FD\approx 28.5](https://tex.z-dn.net/?f=FD%5Capprox%2028.5)
Step-by-step explanation:
Please find the attachment.
We have been given that angle FED with angle F=90 degree, ED=36, and FE=22. We are asked to find the unknown angles and the unknown side length of the triangle.
We will use sine to solve for angle D as:
![\text{sin}=\frac{\text{Opposite}}{\text{Hypotenuse}}](https://tex.z-dn.net/?f=%5Ctext%7Bsin%7D%3D%5Cfrac%7B%5Ctext%7BOpposite%7D%7D%7B%5Ctext%7BHypotenuse%7D%7D)
![\text{sin}(D)=\frac{22}{36}](https://tex.z-dn.net/?f=%5Ctext%7Bsin%7D%28D%29%3D%5Cfrac%7B22%7D%7B36%7D)
![D=\text{sin}^{-1}(\frac{22}{36})](https://tex.z-dn.net/?f=D%3D%5Ctext%7Bsin%7D%5E%7B-1%7D%28%5Cfrac%7B22%7D%7B36%7D%29)
![D=37.66988696^{\circ}](https://tex.z-dn.net/?f=D%3D37.66988696%5E%7B%5Ccirc%7D)
![D\approx 37.7^{\circ}](https://tex.z-dn.net/?f=D%5Capprox%2037.7%5E%7B%5Ccirc%7D)
Therefore, measure of angle D is 37.7 degrees.
Now, we will find measure of angle E using angle sum property.
![m\angle E+m\angle F+m\angle D=180^{\circ}](https://tex.z-dn.net/?f=m%5Cangle%20E%2Bm%5Cangle%20F%2Bm%5Cangle%20D%3D180%5E%7B%5Ccirc%7D)
![m\angle E+90^{\circ}+37.7^{\circ}=180^{\circ}](https://tex.z-dn.net/?f=m%5Cangle%20E%2B90%5E%7B%5Ccirc%7D%2B37.7%5E%7B%5Ccirc%7D%3D180%5E%7B%5Ccirc%7D)
![m\angle E+127.7^{\circ}=180^{\circ}](https://tex.z-dn.net/?f=m%5Cangle%20E%2B127.7%5E%7B%5Ccirc%7D%3D180%5E%7B%5Ccirc%7D)
![m\angle E+127.7^{\circ}-127.7^{\circ}=180^{\circ}-127.7^{\circ}](https://tex.z-dn.net/?f=m%5Cangle%20E%2B127.7%5E%7B%5Ccirc%7D-127.7%5E%7B%5Ccirc%7D%3D180%5E%7B%5Ccirc%7D-127.7%5E%7B%5Ccirc%7D)
![m\angle E=52.3^{\circ}](https://tex.z-dn.net/?f=m%5Cangle%20E%3D52.3%5E%7B%5Ccirc%7D)
Therefore, measure of angle E is 52.3 degrees.
We will use Pythagoras theorem to solve for side FD as:
![FD^2+EF^2=ED^2](https://tex.z-dn.net/?f=FD%5E2%2BEF%5E2%3DED%5E2)
![FD^2+22^2=36^2](https://tex.z-dn.net/?f=FD%5E2%2B22%5E2%3D36%5E2)
![FD^2+484=1296](https://tex.z-dn.net/?f=FD%5E2%2B484%3D1296)
![FD^2=1296-484](https://tex.z-dn.net/?f=FD%5E2%3D1296-484)
![FD^2=812](https://tex.z-dn.net/?f=FD%5E2%3D812)
![FD=\sqrt{812}](https://tex.z-dn.net/?f=FD%3D%5Csqrt%7B812%7D)
![FD=28.495613697\\\\FD\approx 28.5](https://tex.z-dn.net/?f=FD%3D28.495613697%5C%5C%5C%5CFD%5Capprox%2028.5)
Therefore, length of side FD is approximately 28.5 units.
The polygon has 11 sides.
The measure each interior angle is 147.272°.
Solution:
The given shape is a regular polygon.
(a) Number of sides of the polygon = 11
The polygon has 11 sides.
(b) To find the measure of each interior angle:
Each interior angle of a regular polygon
![$=\frac{(n-2) \times 180^{\circ}}{n}](https://tex.z-dn.net/?f=%24%3D%5Cfrac%7B%28n-2%29%20%5Ctimes%20180%5E%7B%5Ccirc%7D%7D%7Bn%7D)
![$=\frac{(11-2) \times 180^{\circ}}{11}](https://tex.z-dn.net/?f=%24%3D%5Cfrac%7B%2811-2%29%20%5Ctimes%20180%5E%7B%5Ccirc%7D%7D%7B11%7D)
![$=\frac{9 \times 180^{\circ}}{11}](https://tex.z-dn.net/?f=%24%3D%5Cfrac%7B9%20%5Ctimes%20180%5E%7B%5Ccirc%7D%7D%7B11%7D)
![=147.272^\circ](https://tex.z-dn.net/?f=%3D147.272%5E%5Ccirc)
The measure each interior angle is 147.272°.
It could be anything between 7.905 and 7.9099
For example: 7.908 or something
$216 x 0.08 = $17.28.
Therefore $17.28 was collected for sales tax.
![\displaystyle\int_{x=0}^{x=1}\int_{y=1}^{y=x}\cos y^2\,\mathrm dy\,\mathrm dx](https://tex.z-dn.net/?f=%5Cdisplaystyle%5Cint_%7Bx%3D0%7D%5E%7Bx%3D1%7D%5Cint_%7By%3D1%7D%5E%7By%3Dx%7D%5Ccos%20y%5E2%5C%2C%5Cmathrm%20dy%5C%2C%5Cmathrm%20dx)
Change the order of integration. The region over which you're integrating can be equivalently described by the set of points in the plane,
.
Then the integral becomes
![\displaystyle\int_{y=0}^{y=1}\int_{x=0}^{x=y}\cos y^2\,\mathrm dx\,\mathrm dy=\int_{y=0}^{y=1}y\cos y^2\,\mathrm dy](https://tex.z-dn.net/?f=%5Cdisplaystyle%5Cint_%7By%3D0%7D%5E%7By%3D1%7D%5Cint_%7Bx%3D0%7D%5E%7Bx%3Dy%7D%5Ccos%20y%5E2%5C%2C%5Cmathrm%20dx%5C%2C%5Cmathrm%20dy%3D%5Cint_%7By%3D0%7D%5E%7By%3D1%7Dy%5Ccos%20y%5E2%5C%2C%5Cmathrm%20dy)
Substitute
,
:
![\displaystyle\frac12\int_{z=0}^{z=1}\cos z\,\mathrm dz=\frac12\sin1](https://tex.z-dn.net/?f=%5Cdisplaystyle%5Cfrac12%5Cint_%7Bz%3D0%7D%5E%7Bz%3D1%7D%5Ccos%20z%5C%2C%5Cmathrm%20dz%3D%5Cfrac12%5Csin1)