To solve this problem, we need to use the formula for the area of a rectangle, which is length times width. We can set up the formula as follows:
Area = length * width
Since we know that the width is 7 greater than ⅔ the length, we can represent the width as follows:
width = ⅔ * length + 7
We can substitute this expression for the width in the area formula to get the following:
Area = length * (⅔ * length + 7)
We are given that the area of the rectangle must be 172, so we can set up the following equation:
172 = length * (⅔ * length + 7)
To solve for the length of the rectangle, we can first distribute the length on the right-hand side of the equation, like this:
172 = length * ⅔ * length + 7 * length
We can then combine like terms on the right-hand side of the equation to get the following:
172 = ⅔ * length^2 + 7 * length
To solve for the length, we can use the quadratic formula, which is as follows:
length = (-b +/- sqrt(b^2 - 4ac)) / 2a
In this case, a = ⅔, b = 7, and c = -172. Plugging these values into the quadratic formula, we get the following:
length = (-7 +/- sqrt(49 - 4 * ⅔ * -172)) / 2 * ⅔
Simplifying this expression, we get the following:
length = (-7 +/- sqrt(49 + 576)) / ⅓
We can further simplify this expression by calculating the value of the square root, like this:
length = (-7 +/- sqrt(625)) / ⅓
Simplifying again, we get the following:
length = (-7 +/- 25) / ⅓
To find the final value of the length, we need to calculate the two possible values of the square root, which are 25 and -25. Plugging these values into the equation for the length, we get the following:
length = (-7 + 25) / ⅓ = 18 / ⅓ = 6
length = (-7 - 25) / ⅓ = -32 / ⅓ = -10
Since the length of a rectangle must be positive, we can discard the negative value and only consider the value of 6. This means that the length of the rectangle is 6, and the width is ⅔ * 6 + 7 = 11. Therefore, the dimensions of the rectangle are 6 by 11.