Answer:
5⁴a²
Step-by-step explanation:
(5³a³)÷5a-¹×5-²a²
5³a³÷5a-¹×5-²a²
5³a³÷5¹-²×a-¹+²
5³a³÷5-¹a
5³a³/5-¹a
5³-(-¹)a³-¹
5⁴a²
Step-by-step explanation:
Let x be the length and y be the width of the rectangular plot.
The plot is bounded on one side by a river and on the other three sides by a single-strand electric fence. It means,
x+2y = 1500
x = 1500 - 2y ....(1)
We know that the area of a rectangular plot is given by :
A = xy ....(2)
Put the value of x from equation (1) in (2)
.....(3)
For largest area, differentiate above area equation wrt y.
![\dfrac{dA}{dy}=\dfrac{d}{dy}(1500y-2y^2)\\\\=1500-4y\\\\\text{Put}\ \dfrac{dA}{dy}=0\\\\1500-4y=0\\\\y=\dfrac{1500}{4}\\\\=375](https://tex.z-dn.net/?f=%5Cdfrac%7BdA%7D%7Bdy%7D%3D%5Cdfrac%7Bd%7D%7Bdy%7D%281500y-2y%5E2%29%5C%5C%5C%5C%3D1500-4y%5C%5C%5C%5C%5Ctext%7BPut%7D%5C%20%5Cdfrac%7BdA%7D%7Bdy%7D%3D0%5C%5C%5C%5C1500-4y%3D0%5C%5C%5C%5Cy%3D%5Cdfrac%7B1500%7D%7B4%7D%5C%5C%5C%5C%3D375)
Put the value of y in equation (1).
x = 1500-2(375)
= 750 m
Put the value of y in equation (3).
![A =1500(375)-2(375)^2\\A=281250\ m^2](https://tex.z-dn.net/?f=A%20%3D1500%28375%29-2%28375%29%5E2%5C%5CA%3D281250%5C%20m%5E2)
Hence, the largest area is 281250 m² and its dimensions are 750 m and 375 m.
Points:
Quadrant I: (+, +)
Quadrant II: (-, +)
Quadrant III: (-, -)
Quadrant IV: (+, -)
Thus the point (5,4) is found in Quadrant I.
Answer:
Step-by-step explanation:
abc = 1
We have to prove that,
![\frac{1}{1+a+b^{-1}}+\frac{1}{1+b+c^{-1}}+\frac{1}{1+c+a^{-1}}=1](https://tex.z-dn.net/?f=%5Cfrac%7B1%7D%7B1%2Ba%2Bb%5E%7B-1%7D%7D%2B%5Cfrac%7B1%7D%7B1%2Bb%2Bc%5E%7B-1%7D%7D%2B%5Cfrac%7B1%7D%7B1%2Bc%2Ba%5E%7B-1%7D%7D%3D1)
We take left hand side of the given equation and solve it,
![\frac{1}{1+a+\frac{1}{b}}+\frac{1}{1+b+\frac{1}{c}}+\frac{1}{1+c+\frac{1}{a}}](https://tex.z-dn.net/?f=%5Cfrac%7B1%7D%7B1%2Ba%2B%5Cfrac%7B1%7D%7Bb%7D%7D%2B%5Cfrac%7B1%7D%7B1%2Bb%2B%5Cfrac%7B1%7D%7Bc%7D%7D%2B%5Cfrac%7B1%7D%7B1%2Bc%2B%5Cfrac%7B1%7D%7Ba%7D%7D)
Since, abc = 1,
and c = ![\frac{1}{ab}](https://tex.z-dn.net/?f=%5Cfrac%7B1%7D%7Bab%7D)
By substituting these values in the expression,
![\frac{1}{1+a+\frac{1}{b}}+\frac{1}{1+b+\frac{1}{c}}+\frac{1}{1+c+\frac{1}{a}}=\frac{1}{1+a+\frac{1}{b}}+\frac{1}{1+b+ab}+\frac{1}{1+\frac{1}{ab}+\frac{1}{a}}](https://tex.z-dn.net/?f=%5Cfrac%7B1%7D%7B1%2Ba%2B%5Cfrac%7B1%7D%7Bb%7D%7D%2B%5Cfrac%7B1%7D%7B1%2Bb%2B%5Cfrac%7B1%7D%7Bc%7D%7D%2B%5Cfrac%7B1%7D%7B1%2Bc%2B%5Cfrac%7B1%7D%7Ba%7D%7D%3D%5Cfrac%7B1%7D%7B1%2Ba%2B%5Cfrac%7B1%7D%7Bb%7D%7D%2B%5Cfrac%7B1%7D%7B1%2Bb%2Bab%7D%2B%5Cfrac%7B1%7D%7B1%2B%5Cfrac%7B1%7D%7Bab%7D%2B%5Cfrac%7B1%7D%7Ba%7D%7D)
![=\frac{b}{b+ab+1}+\frac{1}{1+b+ab}+\frac{ab}{ab+1+b}](https://tex.z-dn.net/?f=%3D%5Cfrac%7Bb%7D%7Bb%2Bab%2B1%7D%2B%5Cfrac%7B1%7D%7B1%2Bb%2Bab%7D%2B%5Cfrac%7Bab%7D%7Bab%2B1%2Bb%7D)
![=\frac{1+b+ab}{1+b+ab}](https://tex.z-dn.net/?f=%3D%5Cfrac%7B1%2Bb%2Bab%7D%7B1%2Bb%2Bab%7D)
![=1](https://tex.z-dn.net/?f=%3D1)
Which equal to the right hand side of the equation.
Hence, ![\frac{1}{1+a+b^{-1}}+\frac{1}{1+b+c^{-1}}+\frac{1}{1+c+a^{-1}}=1](https://tex.z-dn.net/?f=%5Cfrac%7B1%7D%7B1%2Ba%2Bb%5E%7B-1%7D%7D%2B%5Cfrac%7B1%7D%7B1%2Bb%2Bc%5E%7B-1%7D%7D%2B%5Cfrac%7B1%7D%7B1%2Bc%2Ba%5E%7B-1%7D%7D%3D1)