Question

How do you find the perimeter and area of a rectangle with width of `2sqrt7-2sqrt5` and length of `3sqrt7+3sqrt5`?

Answer 1

Perimeter `P = 10 sqrt(7) + 2 sqrt(5)`

Area `A = 12`

Explanation:

The perimeter `P` of a rectangle is the sum of all side lengths.

The formula is `P = 2(l + w)`, where `l` is the length and `w` is the width:

`Rightarrow P = 2((3 sqrt(7) + 3 sqrt(5)) + (2 sqrt(7) - 2 sqrt(5)))`

`Rightarrow P = 2(3 sqrt(7) + 2 sqrt(7) + 3 sqrt(5) - 2 sqrt(5))`

`Rightarrow P = 2(5 sqrt(7) + sqrt(5))`

`therefore P = 10 sqrt(7) + 2 sqrt(5)`

The area `A` of a rectangle is the product of the length and the width.

The formula is `A = l w`:

`Rightarrow A = (3 sqrt(7) + 3 sqrt(5)) times (2 sqrt(7) - 2 sqrt(5))`

`Rightarrow A = 3 sqrt (7) times 2 sqrt(7) + 3 sqrt(7) times (- 2 sqrt(5)) + 3 sqrt(5) times 2 sqrt(7) + 3 sqrt(5) times (- 2 sqrt(5))`

`Rightarrow A = 6 times 7 - 6 sqrt(35) + 6 sqrt(35) - 6 times 5`

`Rightarrow A = 42 - 30`

`therefore A = 12`

Therefore, the perimeter is `10 sqrt(7) + 2 sqrt(5)` and the area is `12`.

Answer 2

Perimeter of rectangle is `2(5sqrt 7+sqrt5)` unit.

Area of rectangle is `12` sq.unit.

Explanation:

Length of rectangle is `l= 3 sqrt7 +3 sqrt 5`
Width of rectangle is `w= 2 sqrt7 -2 sqrt 5`

Perimeter of rectangle is `P=2l+ 2w = 2(3 sqrt7 +3 sqrt 5) +2(2 sqrt7 -2 sqrt 5)`

`P= sqrt7(6+4) + sqrt5 (6-4) = 10sqrt 7 +2sqrt5 =2(5sqrt 7+sqrt5)`unit

Area of rectangle is `A= l*w= (3 sqrt7 +3 sqrt 5)*(2 sqrt7 -2 sqrt 5)`

`A= 6 * 7 - cancel(6 * sqrt 35) + cancel(6 sqrt35) -6*5 = 42-30 =12` sq.unit.

Perimeter of rectangle is `2(5sqrt 7+sqrt5)` unit
Area of rectangle is `12` sq.unit. [Ans]

Answer 3

12

Explanation:

we know area of rectangle = length*width sq unit.

So, `color(green)(Area)` = `[3sqrt7 +3sqrt5]xx[2sqrt7-2sqrt5]`

`rArr 3[sqrt7+sqrt5]xx2[sqrt7-sqrt5]`

`rArr 3xx2xx[(sqrt7)^2-(sqrt5)^2]`

`rArr 3xx2xx[7-5] = 3xx2xx2 =12` sq unit

`color(red)(perimeter)`= 2*[length+width]unit

`rArr 2[{3sqrt7+3sqrt5}+{2sqrt7-2sqrt5}]`

`rArr2[3sqrt7+3sqrt5+2sqrt7-2sqrt5]`

`rArr2[5sqrt7+sqrt5]`

Answer 4

Area`color(blue)(=12 color(white)(a)`square units,`color(blue)(P=2(5sqrt7+sqrt5)`

Explanation:

`W=2sqrt7-2sqrt5`

`L=3sqrt7+3sqrt5`

perimeter of a rectangle`=P=L+W+L+W`

`P=2L+2W=2(L+W)`

`P=2[(3sqrt7+3sqrt5)+(2sqrt7-2sqrt5)]`

`P=2[3sqrt7+3sqrt5+2sqrt7-2sqrt5]`

`P=2[5sqrt7+sqrt5]`

Area`=A=W xx L`

`A=(2sqrt7-2sqrt5) xx (3sqrt7+3sqrt5)`

`color(white)(aaaaaaaaaaaaa)` `2sqrt7-2sqrt5`
`color(white)(aaaaaaaaaa)` `xx underline(3sqrt7+3sqrt5)`
`color(white)(aaaaaaaaaaaaa)` `6 xx 7-6sqrt5sqrt7`
`color(white)(aaaaaaaaaaaaaaaaaaaa)` `6sqrt5sqrt7-6 xx 5`
`color(white)(aaaaaaaaaaaaaa)` `overline(42-30color(blue)(=12`square units