Question

A rectangle has a length of 17 inches less than 7 times its width. If the area of the rectangle is 2204 square inches, find the length of the rectangle.

Answer 1

The length of the rectangle is 200 inches when its area is 2204 square inches with a width of 31 inches.

Let's denote the width of the rectangle as
`\(w\)` inches. According to the given information, the length of the rectangle is
`\(7w - 17\)`inches because it's 17 inches less than 7 times its width.

The formula for the area of a rectangle is length multiplied by width. So, the area
`(\(A\))` of this rectangle can be expressed as the product of its length and width:

`\[A = \text{Length} * \text{Width}\]`

Substituting the given area
`(\(2204\) square inches)` into the formula, we get:

`\[2204 = (7w - 17) * w\]`

Expanding the equation:

`\[2204 = 7w^2 - 17w\]`

Now, rearrange the equation into a quadratic form:

`\[7w^2 - 17w - 2204 = 0\]`

To solve for w (the width), use quadratic formula or factorization. After solving, we find that w = 31 inches (ignoring the negative solution as width can't be negative).

Substitute w = 31into the expression for the length:

Length
`\(= 7w - 17 = 7 * 31 - 17 = 217 - 17 = 200\)` inches.

Therefore, the length of the rectangle is 200 inches.