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:
the answer is -77/8 or -9.625
Step-by-step explanation:
70%, because 70/100 = .70 which is equal to 70%.
Multiply the numbers (2*2=4)
![=\frac{-7+\sqrt{7^2-4\cdot \:2\left(-5\right)}}{2\cdot \:2}](https://tex.z-dn.net/?f=%3D%5Cfrac%7B-7%2B%5Csqrt%7B7%5E2-4%5Ccdot%20%5C%3A2%5Cleft%28-5%5Cright%29%7D%7D%7B2%5Ccdot%20%5C%3A2%7D)
Apply the rule
![\sqrt{7^2+2\cdot \:4\cdot \:5}](https://tex.z-dn.net/?f=%5Csqrt%7B7%5E2%2B2%5Ccdot%20%5C%3A4%5Ccdot%20%5C%3A5%7D)
7^2 =49
![=\sqrt{49+2\cdot \:4\cdot \:5}](https://tex.z-dn.net/?f=%3D%5Csqrt%7B49%2B2%5Ccdot%20%5C%3A4%5Ccdot%20%5C%3A5%7D)
Multiply
![\sqrt{49+40}](https://tex.z-dn.net/?f=%5Csqrt%7B49%2B40%7D)
Add
![\sqrt{89}](https://tex.z-dn.net/?f=%5Csqrt%7B89%7D)
![=\frac{-7-\sqrt{89}}{4}](https://tex.z-dn.net/?f=%3D%5Cfrac%7B-7-%5Csqrt%7B89%7D%7D%7B4%7D)
Final solution: