The area of the region that lies inside both curves r² = 2sin(2θ), r = 1 is -0.1287 square units.
<h3>How to find the area of the region that lies inside both curves?</h3>
Since the curves are
We find their point of intersection.
So, r² = r²
2sin(2θ) = 1²
2sin(2θ) = 1
sin(2θ) = 1/2
2θ = sin⁻¹(1/2)
2θ = π/6
θ = π/12
So, we integrate the area from θ = 0 to θ = π/12
Now the area A of the region between two curves between θ = α to θ = β is
![A = \int\limits^{\beta }_\alpha ({r^{2} - r^{'2} }) \, d\theta](https://tex.z-dn.net/?f=A%20%3D%20%5Cint%5Climits%5E%7B%5Cbeta%20%7D_%5Calpha%20%20%28%7Br%5E%7B2%7D%20-%20r%5E%7B%272%7D%20%7D%29%20%5C%2C%20d%5Ctheta)
So, the area betwwen the curves r² = 2sin(2θ), r = 1 between θ = 0 to θ = π/12 is
![A = \int\limits^{\frac{\pi}{12} }_0 ({r^{2} - r^{'2} }) \, d\theta \\= \int\limits^{\frac{\pi}{12} }_0 ({2sin2\theta - 1^{2} }) \, d\theta\\= \int\limits^{\frac{\pi}{12} }_0 ({2sin2\theta - 1}) \, d\theta\\= \int\limits^{\frac{\pi}{12} }_0 {2sin2\theta}\, d\theta - \int\limits^{\frac{\pi}{12} }_0 {1} \, d\theta\\= [2(-cos2\theta)/2 - \theta]_{0}^{\frac{\pi}{12} } \\= [-cos2\theta - \theta]_{0}^{\frac{\pi}{12} } \\= -cos2\frac{\pi}{12} - \frac{\pi}{12} - ( -cos2(0) - 0)\\](https://tex.z-dn.net/?f=A%20%3D%20%5Cint%5Climits%5E%7B%5Cfrac%7B%5Cpi%7D%7B12%7D%20%7D_0%20%28%7Br%5E%7B2%7D%20-%20r%5E%7B%272%7D%20%7D%29%20%5C%2C%20d%5Ctheta%20%5C%5C%3D%20%20%5Cint%5Climits%5E%7B%5Cfrac%7B%5Cpi%7D%7B12%7D%20%7D_0%20%28%7B2sin2%5Ctheta%20-%201%5E%7B2%7D%20%7D%29%20%5C%2C%20d%5Ctheta%5C%5C%3D%20%20%5Cint%5Climits%5E%7B%5Cfrac%7B%5Cpi%7D%7B12%7D%20%7D_0%20%28%7B2sin2%5Ctheta%20-%201%7D%29%20%5C%2C%20d%5Ctheta%5C%5C%3D%20%20%5Cint%5Climits%5E%7B%5Cfrac%7B%5Cpi%7D%7B12%7D%20%7D_0%20%7B2sin2%5Ctheta%7D%5C%2C%20d%5Ctheta%20-%20%20%5Cint%5Climits%5E%7B%5Cfrac%7B%5Cpi%7D%7B12%7D%20%7D_0%20%7B1%7D%20%5C%2C%20d%5Ctheta%5C%5C%3D%20%5B2%28-cos2%5Ctheta%29%2F2%20-%20%5Ctheta%5D_%7B0%7D%5E%7B%5Cfrac%7B%5Cpi%7D%7B12%7D%20%7D%20%5C%5C%3D%20%5B-cos2%5Ctheta%20-%20%5Ctheta%5D_%7B0%7D%5E%7B%5Cfrac%7B%5Cpi%7D%7B12%7D%20%7D%20%5C%5C%3D%20-cos2%5Cfrac%7B%5Cpi%7D%7B12%7D%20-%20%5Cfrac%7B%5Cpi%7D%7B12%7D%20%20-%20%28%20-cos2%280%29%20-%200%29%5C%5C)
![= -cos2(\frac{\pi}{12}) - \frac{\pi}{12} - ( -cos2(0) - 0)\\= -cos\frac{\pi}{6}- \frac{\pi}{12} - ( -cos0 - 0)\\= -cos\frac{\pi}{6}- \frac{\pi}{12} - ( -1)\\= -(\frac{\sqrt{3} }{2} ) - \frac{\pi}{12} + 1\\= -0.8660 - 0.2618 + 1\\= -1.1278 + 1\\= -0.1278](https://tex.z-dn.net/?f=%3D%20-cos2%28%5Cfrac%7B%5Cpi%7D%7B12%7D%29%20-%20%5Cfrac%7B%5Cpi%7D%7B12%7D%20%20-%20%28%20-cos2%280%29%20-%200%29%5C%5C%3D%20-cos%5Cfrac%7B%5Cpi%7D%7B6%7D-%20%5Cfrac%7B%5Cpi%7D%7B12%7D%20%20-%20%28%20-cos0%20-%200%29%5C%5C%3D%20-cos%5Cfrac%7B%5Cpi%7D%7B6%7D-%20%5Cfrac%7B%5Cpi%7D%7B12%7D%20%20-%20%28%20-1%29%5C%5C%3D%20-%28%5Cfrac%7B%5Csqrt%7B3%7D%20%7D%7B2%7D%20%29%20-%20%5Cfrac%7B%5Cpi%7D%7B12%7D%20%20%2B%201%5C%5C%3D%20-0.8660%20-%200.2618%20%2B%201%5C%5C%3D%20-1.1278%20%2B%201%5C%5C%3D%20-0.1278)
So, the area of the region that lies inside both curves r² = 2sin(2θ), r = 1 is -0.1287 square units.
Learn more about area of region between curves here:
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You set it up as fractions 4/5 = 28/x and solve for x then add the numbers together. 5*28=140 140/4= 35 35+28= 63
Answer:
17. C 60°
18. D 121°
Step-by-step explanation:
The external angle is half the difference of the intercepted arcs.
17. (161° -41°)/2 = 60°
__
18. (? -59°)/2 = 31°
? = 59° +2(31°) = 121°
Answer:
x=-4,1,2+5i,2-5i
Step-by-step explanation:
Given is an algebraic expression g(x) as product of two functions.
Hence solutions will be the combined solutions of two quadratic products
![g(x) = (x^2 + 3x - 4)(x^2 - 4x + 29)\\](https://tex.z-dn.net/?f=g%28x%29%20%3D%20%28x%5E2%20%2B%203x%20-%204%29%28x%5E2%20-%204x%20%2B%2029%29%5C%5C)
I expression can be factorised as
![(x+4)(x-1)](https://tex.z-dn.net/?f=%28x%2B4%29%28x-1%29)
Hence one set of solutions are
x=-4,1
Next quadratic we cannot factorize
and hence use formulae
![x=\frac{4+/-\sqrt{16-116} }{2} =2+5i, 2-5i](https://tex.z-dn.net/?f=x%3D%5Cfrac%7B4%2B%2F-%5Csqrt%7B16-116%7D%20%7D%7B2%7D%20%3D2%2B5i%2C%202-5i)
Find 33% of 185:
0.33 * 185 = 61.05
Subtract this from the regular price:
185 - 61.05 = 123.95
So the sale price is $123.95