Answer:
The answer is below
Step-by-step explanation:
Find the dimensions of the rectangle of maximum area with sides parallel to the coordinate axes that can be inscribed in the ellipse 4x² + 16y² = 16
Solution:
Given that the ellipse has the equation: 4x² + 16y² = 16
let us make x the subject of the formula, hence:
4x² + 16y² = 16
4x² = 16 - 16y²
Dividing through by 4:
x² = (16 - 16y²)/4
x² = 4 - 4y²
Taking square root of both sides:
![x=\sqrt{4-4y^2 }\\](https://tex.z-dn.net/?f=x%3D%5Csqrt%7B4-4y%5E2%20%7D%5C%5C)
The points of the rectangle vertices is at (x,y), (-x,y), (x,-y), (-x,-y). Hence the rectangle has length and width of 2x and 2y.
The area of a rectangle inscribed inside an ellipse is given by:
Area (A) = 4xy
A = 4xy
![A=4(\sqrt{4-4y^2} )y\\\\A=4y\sqrt{4-4y^2}=4\sqrt{4y^2-4y^4} \\\\The\ maximum\ area\ of\ the\ rectangle\ is\ at\ \frac{dA}{dy}=0\\\\ \frac{dA}{dy}=4(\frac{4-8y^2}{\sqrt{4-4y^2} } )\\\\4(\frac{4-8y^2}{\sqrt{4-4y^2} } )=0\\\\4-8y^2=0\\\\8y^2=4\\\\y^2=1/2\\\\y=\frac{1}{\sqrt{2} }\\\\x=\sqrt{4-4(\frac{1}{\sqrt{2} })^2}=\sqrt{2}](https://tex.z-dn.net/?f=A%3D4%28%5Csqrt%7B4-4y%5E2%7D%20%29y%5C%5C%5C%5CA%3D4y%5Csqrt%7B4-4y%5E2%7D%3D4%5Csqrt%7B4y%5E2-4y%5E4%7D%20%20%5C%5C%5C%5CThe%5C%20maximum%5C%20area%5C%20of%5C%20the%5C%20rectangle%5C%20is%5C%20at%5C%20%5Cfrac%7BdA%7D%7Bdy%7D%3D0%5C%5C%5C%5C%20%20%5Cfrac%7BdA%7D%7Bdy%7D%3D4%28%5Cfrac%7B4-8y%5E2%7D%7B%5Csqrt%7B4-4y%5E2%7D%20%7D%20%29%5C%5C%5C%5C4%28%5Cfrac%7B4-8y%5E2%7D%7B%5Csqrt%7B4-4y%5E2%7D%20%7D%20%29%3D0%5C%5C%5C%5C4-8y%5E2%3D0%5C%5C%5C%5C8y%5E2%3D4%5C%5C%5C%5Cy%5E2%3D1%2F2%5C%5C%5C%5Cy%3D%5Cfrac%7B1%7D%7B%5Csqrt%7B2%7D%20%7D%5C%5C%5C%5Cx%3D%5Csqrt%7B4-4%28%5Cfrac%7B1%7D%7B%5Csqrt%7B2%7D%20%7D%29%5E2%7D%3D%5Csqrt%7B2%7D)
Therefore the length = 2x = 2√2, the width = 2y = 2/√2
Answer:
51.3
Step-by-step explanation:
This is a right triangle (because of the 90° angle). For any right triangle, you can use the following:
Sine = Opposite / Hypotenuse
Cosine = Adjacent / Hypotenuse
Tangent = Opposite / Adjacent
You can remember this as SOH-CAH-TOA.
We want to find the angle y. We know that the opposite side is 10 and the adjacent side is 8. So we can use tangent:
tan y = 10 / 8
y ≈ 51.3°
1 Pint= 2 Cups= 16 ounces
1 Quart= 2 pints= 4 cups= 32 ounces
The scientific notation:
![a\cdot10^n\\\\1\leq a < 10;\ n\in\mathbb{Z}](https://tex.z-dn.net/?f=a%5Ccdot10%5En%5C%5C%5C%5C1%5Cleq%20a%20%3C%2010%3B%5C%20n%5Cin%5Cmathbb%7BZ%7D)
![(4\cdot10^8)^2=4^2(10^8)^2=16\cdot10^{8\cdot2}=16\cdot10^{16}\\\\=1.6\cdot10\cdot10^{16}=1.6\cdot10^{17}](https://tex.z-dn.net/?f=%284%5Ccdot10%5E8%29%5E2%3D4%5E2%2810%5E8%29%5E2%3D16%5Ccdot10%5E%7B8%5Ccdot2%7D%3D16%5Ccdot10%5E%7B16%7D%5C%5C%5C%5C%3D1.6%5Ccdot10%5Ccdot10%5E%7B16%7D%3D1.6%5Ccdot10%5E%7B17%7D)
Used:
![(a\cdot b)^n=a^n\cdot b^n\\\\(a^n)^m=a^{n\cdot m}\\\\a^n\cdot a=a^{n+1}](https://tex.z-dn.net/?f=%28a%5Ccdot%20b%29%5En%3Da%5En%5Ccdot%20b%5En%5C%5C%5C%5C%28a%5En%29%5Em%3Da%5E%7Bn%5Ccdot%20m%7D%5C%5C%5C%5Ca%5En%5Ccdot%20a%3Da%5E%7Bn%2B1%7D)
Answer: (4 × 10⁸)² = 1.6 × 10¹⁷
Answer:
An insurance company polls drivers of sports cars to find their average number of tickets.
Step-by-step explanation:
It’s bias because they’re only polling people with sports cars