Answer:
the area of the parallelogram = base times the height
For the given parallelogram :
x = 8 units, y = 13 units, and h = 11 units
1) If x is the perpendicular to h
So, the base = 8 and the height = 11
So, the area = 8 * 11 = 88 square units
2) If y is the perpendicular to h
So, the base = 13 and the height = 11
So, the area = 13 * 11 = 143 square units
Step-by-step explanation:
Answer:
![$y(t) = C_1 e^{t} + C_2 e^{-t} + D_1 \cos t + D_2 \sin t }$](https://tex.z-dn.net/?f=%20%24y%28t%29%20%3D%20C_1%20e%5E%7Bt%7D%20%2B%20C_2%20e%5E%7B-t%7D%20%2B%20D_1%20%5Ccos%20t%20%2B%20D_2%20%5Csin%20t%20%7D%24%20)
Step-by-step explanation:
The equation is a<em> </em><em>linear differential equation: y⁽⁴⁾- y = 0 </em>
We assume the form of the solution y(t) is ![$y(t)=C_{1} e^{\alpha_{1} t} + C_{2} e^{\alpha_{2} t} + C_{3} e^{\alpha_{3} t} + C_{4} e^{\alpha_{4} t} $](https://tex.z-dn.net/?f=%24y%28t%29%3DC_%7B1%7D%20e%5E%7B%5Calpha_%7B1%7D%20t%7D%20%2B%20C_%7B2%7D%20e%5E%7B%5Calpha_%7B2%7D%20t%7D%20%2B%20C_%7B3%7D%20e%5E%7B%5Calpha_%7B3%7D%20t%7D%20%2B%20C_%7B4%7D%20e%5E%7B%5Calpha_%7B4%7D%20t%7D%20%24)
where
are the roots of the auxiliary equation.
So, use the auxiliary equation:
to find the roots; the values are : α₁ = 1, α₂ = -1, α₃ = i, α₄ = -i
Then inserting
values in the assumed solution
⇒ <em>
</em>
Also, because the last 2 terms have complex power, the solution can be written with cosine and sine terms:
<em>Using the Euler's formula:
, we can rewrite the solution as:</em>
= ![C_{1} e^{t} + C_{2} e^{-t} + C_{3} ( \cos t + i \sin t ) + C_{4} ( \cos t - i \sin t ) = C_{1} e^{t} + C_{2} e^{-t} + \cos t ( C_{3} + C_{4} ) + \sin t (i C_{3} - i C_{4} ) = C_{1} e^{t} + C_{2} e^{-t} + D_{1} \cos t +D_{2} \sin t$](https://tex.z-dn.net/?f=C_%7B1%7D%20e%5E%7Bt%7D%20%2B%20C_%7B2%7D%20e%5E%7B-t%7D%20%2B%20C_%7B3%7D%20%28%20%5Ccos%20t%20%2B%20i%20%5Csin%20t%20%29%20%2B%20C_%7B4%7D%20%28%20%5Ccos%20t%20-%20i%20%5Csin%20t%20%29%20%3D%20C_%7B1%7D%20e%5E%7Bt%7D%20%2B%20C_%7B2%7D%20e%5E%7B-t%7D%20%2B%20%5Ccos%20t%20%28%20C_%7B3%7D%20%2B%20C_%7B4%7D%20%29%20%2B%20%5Csin%20t%20%28i%20C_%7B3%7D%20-%20i%20C_%7B4%7D%20%29%20%3D%20C_%7B1%7D%20e%5E%7Bt%7D%20%2B%20C_%7B2%7D%20e%5E%7B-t%7D%20%2B%20D_%7B1%7D%20%5Ccos%20t%20%2BD_%7B2%7D%20%5Csin%20t%24%20)
<em>Where: </em>![$D_1 = C_3 + C_4$ and $D_2= i ( C_3 - C_4 )$](https://tex.z-dn.net/?f=%20%24D_1%20%3D%20C_3%20%2B%20C_4%24%20and%20%24D_2%3D%20i%20%28%20C_3%20-%20C_4%20%29%24%20)
<em>Finally the solution for de linear differential equation y^(4) - y =0 is:</em>
![$y(t) = C_1 e^{t} + C_2 e^{-t} + D_1 \cos t + D_2 \sin t }$](https://tex.z-dn.net/?f=%20%24y%28t%29%20%3D%20C_1%20e%5E%7Bt%7D%20%2B%20C_2%20e%5E%7B-t%7D%20%2B%20D_1%20%5Ccos%20t%20%2B%20D_2%20%5Csin%20t%20%7D%24%20)
<em> </em>
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The length of the room in inches is 126 inches.