Is it possible to construct a rectangle whose length is 1cm less than twice its width, and whose area is 40cm^2? If your answer is yes, determine the dimensions of the rectangle to the nearest tenth of a centimetre. If your answer is no, explain why it is not possible.

I know it's possible, but I don't know how to get the answer. Please help me! My test is tomorrow. ><

Thanks!

I'm at a loss, so I put impossible rectangle in the search engine. A site called Stella's Stunner's may be able to help, good luck!

Perimetetr

To find the dimensions of the rectangle, we can use algebraic equations.

Let's assume the width of the rectangle is "w" cm. According to the problem, the length is 1 cm less than twice the width, which can be expressed as (2w - 1) cm.

Now, we can use the formula for the area of a rectangle, which is length multiplied by width, to form an equation:

Area = Length × Width
40 cm^2 = (2w - 1) cm × w cm

Now, let's simplify the equation:

40 = 2w^2 - w

Rearranging the equation will give us a quadratic equation:

2w^2 - w - 40 = 0

To solve for "w," we can use factoring, completing the square, or the quadratic formula. Let's use the quadratic formula, which states that for an equation of the form ax^2 + bx + c = 0, the solutions for x are given by:

x = (-b ± √(b^2 - 4ac)) / 2a

In our case, a = 2, b = -1, and c = -40. Substituting these values into the quadratic formula, we get:

w = (-(-1) ± √((-1)^2 - 4(2)(-40))) / (2(2))

Simplifying further:

w = (1 ± √(1 + 320)) / 4

w = (1 ± √321) / 4

Therefore, the width of the rectangle can be found by evaluating this equation for both the positive and negative roots of √321. Let's calculate this using a calculator:

w ≈ (1 + √321) / 4 ≈ 6.22 cm (rounded to the nearest tenth)

or

w ≈ (1 - √321) / 4 ≈ -5.72 cm (rounded to the nearest tenth)

Since a negative value for the width does not make sense in the context of a physical rectangle, we discard the negative value.

So, the width of the rectangle is approximately 6.22 cm. To find the length, we substitute this value back into the expression for the length:

Length = 2w - 1
Length ≈ 2(6.22) - 1 ≈ 12.43 - 1 ≈ 11.43 cm (rounded to the nearest tenth)

Therefore, the dimensions of the rectangle to the nearest tenth of a centimeter are approximately 6.22 cm for the width and 11.43 cm for the length.