Question

We are interested in the dimensions of a certain square. A rectangle has length twice the side of this square and width three units less than the side of this square. If the two areas are equal, what are the square's dimensions (w x h)?

Answer 1

To find the dimensions of the square, set up a proportion using the given information and solve the equation.

Step-by-step explanation:

To find the dimensions of the square, we can set up a proportion using the given information.

The length of the rectangle is twice the side length of the square, so it would be 2x.

The width of the rectangle is three units less than the side length of the square, so it would be x-3.

We can set up the equation: 2x(x-3) = x^2.

Simplifying this equation, we get: 2x^2 - 6x = x^2.

Moving all terms to one side, we have: x^2 - 6x = 0.

Factoring out x, we get: x(x - 6) = 0. So, x = 0 or x = 6. Since the side length of the square cannot be zero, the dimensions of the square are 6 x 6.

Answer 2

We don't need the length AND width of the square. One number
will be enough for an answer, because all sides of a square have
the same length. Let's call that number ' M ' (for 'mystery number').

-- The area of the square is M² .

The question says that ...
-- The length of the rectangle is 2M .
-- The width of the rectangle is (M - 3) .

Therefore ...
-- The area of the rectangle is (2M)(M - 3) = 2M² - 6M .

The question also says that ...
-- The areas of the square and the rectangle are equal:

So 2M² - 6M = M²

Subtract M² from each side: M² - 6M = 0

Divide each side by M : M - 6 = 0

Add 6 to each side : M = 6 .


Check:

Assume ... Side of the square = 6

Then ...
Area of the square = 6² = 36 .
Length of the rectangle = (6 x 2) = 12
Width of the rectangle = (6 - 3) = 3
Area of the rectangle = (12 x 3) = 36

36 = 36 yay!