The length of the curve (and thus the total distance traveled by the particle along the curve) is
![\displaystyle\int_0^{3\pi}\sqrt{x'(t)^2+y'(t)^2}\,\mathrm dt](https://tex.z-dn.net/?f=%5Cdisplaystyle%5Cint_0%5E%7B3%5Cpi%7D%5Csqrt%7Bx%27%28t%29%5E2%2By%27%28t%29%5E2%7D%5C%2C%5Cmathrm%20dt)
We have
<em>x(t)</em> = 3 sin²(<em>t</em> ) ==> <em>x'(t)</em> = 6 sin(<em>t</em> ) cos(<em>t</em> ) = 3 sin(2<em>t</em> )
<em>y(t)</em> = 3 cos²(<em>t</em> ) ==> <em>y'(t)</em> = -6 cos(<em>t</em> ) sin(<em>t</em> ) = -3 sin(2<em>t</em> )
Then
√(<em>x'(t)</em> ² + <em>y'(t)</em> ²) = √(18 sin²(2<em>t</em> )) = 18 |sin(2<em>t</em> )|
and the arc length is
![\displaystyle 18 \int_0^{3\pi} |\sin(2t)| \,\mathrm dt](https://tex.z-dn.net/?f=%5Cdisplaystyle%2018%20%5Cint_0%5E%7B3%5Cpi%7D%20%7C%5Csin%282t%29%7C%20%5C%2C%5Cmathrm%20dt)
Recall the definition of absolute value: |<em>x</em>| = <em>x</em> if <em>x</em> ≥ 0, and |<em>x</em>| = -<em>x</em> if <em>x</em> < 0.
Now,
• sin(2<em>t</em> ) ≥ 0 for <em>t</em> ∈ (0, <em>π</em>/2) U (<em>π</em>, 3<em>π</em>/2) U (2<em>π</em>, 5<em>π</em>/2)
• sin(2<em>t</em> ) < 0 for <em>t</em> ∈ (<em>π</em>/2, <em>π</em>) U (3<em>π</em>/2, 2<em>π</em>) U (5<em>π</em>/2, 3<em>π</em>)
so we split up the integral as
![\displaystyle 18 \left(\int_0^{\pi/2} \sin(2t) \,\mathrm dt - \int_{\pi/2}^\pi \sin(2t) \,\mathrm dt + \cdots - \int_{5\pi/2}^{3\pi} \sin(2t) \,\mathrm dt\right)](https://tex.z-dn.net/?f=%5Cdisplaystyle%2018%20%5Cleft%28%5Cint_0%5E%7B%5Cpi%2F2%7D%20%5Csin%282t%29%20%5C%2C%5Cmathrm%20dt%20-%20%5Cint_%7B%5Cpi%2F2%7D%5E%5Cpi%20%5Csin%282t%29%20%5C%2C%5Cmathrm%20dt%20%2B%20%5Ccdots%20-%20%5Cint_%7B5%5Cpi%2F2%7D%5E%7B3%5Cpi%7D%20%5Csin%282t%29%20%5C%2C%5Cmathrm%20dt%5Cright%29)
which evaluates to 18 × (1 - (-1) + 1 - (-1) + 1 - (-1)) = 18 × 6 = 108.
Answer:
Area of green trapezoid would be:
(length of two parallel sides × h) ÷ 2
( 5cm x 2cm) ÷ 2
= 5 square centimetres
Area of the orange trapezoid would be:
(length of two parallel sides x h) ÷ 2
(10.5 cm × 3) ÷ 2
( 31. 5) ÷ 2
(15. 75 square centimetres
NOTE: h represents height
1/3 of the original number is X.
<span>Therefore, the original number is 3X </span>
<span>3X + X = 1348 </span>
<span>4X = 1348 </span>
<span>X = 1348/4 </span>
<span>3X = 3(1348/4)
x=1011</span>
If the denominator is 0 then it is undefined slope
You need to rearrange the problem so that you get Y by it self....
subtract 4x to both sides and you are left with 3y=-4x+12, now divide both sides by 3 and you get y=-4/3x+4 slope= -4/3