Question

the length of a rectangle is 9 more than the width. the area is 112 square yards. find the length and width of the rectangle.

Answer 1

length: 16

width: 7

Explanation:

we are given the length of a rectangle is 9 more than the width. let the width be w. length will be
`w+9`.

so, we have area as 112. this gives us the quadratic equation
`w(w+9)=112`

expanding and simplifying, we get

`w^2+9w=112\implies w^2+9w-112=0`.

next, use the quadratic formula to find w.

find the discriminant D first. (recall
`D=b^2-4ac`)

`D=9^2-4(1)(-112)=529`

next, we will use the quadratic formula. note that
`√(529)=23`.

`w=(-9\pm23)/(2)`

from here we get
`w=(14)/(2)=7.` ignore the negative solution since width can't be negative.

so, now that we have width, the length will be
`l=7+9=16.`

final test,
`7*16=112.` correct.

Answer 2

`\sf length = 16` yards

`\sf width = 7` yards

Explanation:

Let's denote the width of the rectangle as
`\sf w` and the length as
`\sf l`. According to the given information:

1. The length is 9 more than the width:
`\sf l = w + 9`.

2. The area of the rectangle is 112 square yards:
`\sf lw = 112`.

Now, we can set up a system of equations using the above information:

`\sf l = w + 9`

`\sf lw = 112`

Substitute the expression for
`\sf l` from the first equation into the second equation:

`\sf (w + 9)w = 112`

Expand and rearrange the equation:

`\sf w^2 + 9w - 112 = 0`

Now, factor the quadratic equation:

`\sf w^2 + (16 -7)w - 112 = 0`

`\sf w^2 + 16w - 7w - 112 = 0`

`\sf w(w+16)-7(w+16)`

`\sf (w - 7)(w + 16) = 0`

This gives two possible solutions for
`\sf w`:

1.
`\sf w - 7 = 0 \implies w = 7`

2.
`\sf w + 16 = 0 \implies w = -16` (discarded as width cannot be negative)

So, the width of the rectangle is
`\sf w = 7`. Now, substitute this value back into the expression for
`\sf l`:

`\sf l = w + 9`

`\sf l = 7 + 9`

`\sf l = 16`

Therefore, the length of the rectangle is
`\sf l = 16` yards, and the width is
`\sf w = 7` yards.