Question

john is creating a rectangular garden in his backyard. the length of the garden is 16 feet. the perimeter of the garden must be at least 70 feet and no more than 112 feet. use a compound inequality to find the range of values for the width w of the garden

Answer 1

Explanation:

Let length = x

Let width = (1/2)x+2

(2*length)+(2*width) = Perimeter

2x+2[(1/2)x+2] = 40

2x+x+4 = 40

3x+4 = 40

3x = 36

x = 12

Substitute this value to find the length and width.

length = 12 feet

width = (12/2)+2 = 8 feet

Lets perform a check

12+12+8+8 = 40

24+16 = 40

40 = 40

Answer 2

We can write the range for w as :
`19\leq w\leq 40`

Explanation:

The perimeter of rectangle is given as :

`P=2(l+w)` or
`P=2l+2w`

Where l = length and w = width

Given is : the length of the garden is 16 feet

Also given is that the perimeter of the garden must be at least 70 feet and no more than 112 feet.

So, this can be shown as :

`70\leq P\leq 112`

=>
`70\leq (2l+2w)\leq 112`

Putting l = 16

=>
`70\leq (2(16)+2w)\leq 112`

=>
`70\leq (32+2w)\leq 112`

=>
`70\leq (32+2w) \leq 112`

=>
`70-32 \leq 2w` and
`2w \leq 112-32`

=>
`38 \leq 2w` and
`2w \leq 80`

=>
`19 \leq w` and
`w \leq 40`

So, we can write the range for w as :
`19\leq w\leq 40`