Question

The area of a rectangle is 27 square meters. If the length is 6 meters less than 3 times the width, then find the dimensions of the rectangle. Round off your answers to the nearest hundredth.?

Answer 1

`\color{blue}{6.487 m, 4.162m}`

Explanation:

Let `L` & `B` be the length & width of rectangle then as per given conditions,

`L=3B-6 \ .........(1)`

`LB=27 \ .........(2)`

substituting the value of L from (1) into (2) as follows

`(3B-6)B=27`

`B^2-2B-9=0`

`B=\frac{-(-2)\pm\sqrt{(-2)^2-4(1)(-9)}}{2(1)}`
`=1\pm\sqrt{10}`

since, `B>0`, hence we get

`B=1+\sqrt{10}` &
`L=3(1+\sqrt{10})-6`
`L=3(\sqrt{10}-1)`

Hence, length & width of given rectangle are

`L=3(\sqrt{10}-1)\approx 6.486832980505138\ m`
`B=\sqrt{10}+1\approx 4.16227766016838\ m`

Answer 2

length = m = 6.49
width = n = 4.16

Explanation:

Assume that length = `m` and width = `n`.
The rectangle's area will hence be `mn`.
The first statement states "The area of a rectangle is 27 square meters.
Hence `mn=27`.
The second statement states "If the length is 6 meters less than 3 times the width..."
Therefore `m=3n-6`
Now you can create a system of equations:
`mn=27`
`m=3n-6`

Replace `m` in the first equation with `3n-6`:
`(3n-6)*n=27`
Expand the bracket:
`3n^2-6*n=27`
Make a quadratic equation:
`3n^2-6*n-27=0`
Simplify by dividing everything by 3:
`n^2-2*n-9=0`
Use `(-b+-sqrt(b^2-4ac))/(2a)`, where `a` is 1, `b` is -2, and `c` is -9:
=`(2+-sqrt(4+36))/(2)`
=`1+-sqrt10`

Since dimensions must be positive, `n` will be `1+sqrt10`, which to the nearest hundredths is 4.16.
Use `mn=27` to find `m`:
`m(1+sqrt10)=27`
`m=27/(1+sqrt10)`
`m=6.49`