Question

How do you solve this?

Answer 1

Width: 7

Length: 14

Explanation:

The area of a rectangle can be found my multiplying the length by the width. We do not know neither the length nor the width. However, we do know that the length is 7 feet longer than the width. Because we do not know the value of either side, let's let
`x` represent the width.

Since the length is 7 feet longer, we can represent the length by
`x+7`.

Now that we have our values for the length and the width we can multiply them together to find our area.

`(x)(x+7)=x^(2) +7x`

Now we have our equation, so we can address the changes made by the original question. The original question states that when 7 is added to both the length and width the area becomes 3 times larger. To do so simply increase each side by 7 by adding 7 to the original values.

`(x+7)(x+14)`

And multiply our area by 3.

`3(x^(2) +7x)`

set these equal to each other to find your new equation.

`(x+7)(x+14)=3(x^(2) +7x)`

Now you need to solve for
`x`. To do this first muliply
`(x+7)` and
`(x+14)`.

`(x+7)(x+14)=\\x^(2) +14x+7x+98=\\x^(2) +21x+98`

Then multiply
`3(x^(2) +7x)`

`3(x^(2) +7x)=\\3x^(2) +21x`

Now you can begin solving for
`x`.

`x^(2) +21x+98=3x^(2) +21x`

Subtract
`x^(2)` from both sides.

`21x+98=2x^(2) +21x`

Subtract
`21x` from both sides.

`98=2x^(2)`

Divide by 2.

`49=x^(2)`

And finally take the sqaure root of both sides.

`\sqrt{x^(2) }=√(49)\\x=7`

Remember, because we are dealing with length, there cannot be negative. So while normally we would get both +7 and -7, in this case we only get +7.

Now that we have the value of
`x`, we can plug it into our original values.

For width we simply get 7.

For length we get 7+7=14

To check our answer we cna multiply 7 by 14 to get 98. Then we can add 7 to our length and width to get 14 and 21. Multiply these together and we get 294. 294 divided by 3 is 98, proving our answer correct.