Question

solve for x
if don't know how to solve
then no answer​

Answer 1

Step-by-step explanation:

First consider the general case with a ladder of length l and the cubical box with a side c.

Construct a square from 3 rotated copies of this diagram as follows:

Connect the four corners of the cubical boxes to form a square with side length w = x + y.

Let X be the area of the triangle with side x and Y the same for y. Write Z for the area of the triangle “between” X and Y.

Now let’s solve for the area of the green square in two different ways.

Since the square has a side length w + c, its area is the square of the side length:

Green area = (w + c)2

We can also calculate the green area as the sum of four rectangles: a square with side length c, two rectangles with sides of x + y and c, and another square with side length w. Adding those areas gives the equation:

Green area = c2 + 2(X + Y) + 2Z + w2

In the rectangle with sides of x + y and c, the quantity Z is half of the rectangle’s area, as is the area X + Y. Thus X + Y = Z, and we have:

Green area = c2 + 2(Z) + 2Z + w2

Green area = c2 + 4Z + w2

We can further simplify because 4Z + w2 is the area of the square with side length l. Hence 4Z + w2 = l2 and we have:

Green area = c2 + l2

Equating the two expressions for the green area gives:

(w + c)2 = c2 + l2

Now we can take the positive square root of both sides (positive since we want a positive length) to get and expression for w in terms of c and l:

w + c = √(c2 + l2)

w = –c + √(c2 + l2)

Now recall w = x + y. Then consider the two similar right triangles formed between the ladder and the box:

Because the triangles are similar we have:

x/c = c/y

x = c2/y

We then have:

w = x + y

w = c2/y + y

wy = c2 + y2

0 = y2 – wy + c2

The above is a quadratic equation in y so we can solve:

y = 0.5(w ± √(w2 – 4c2))

Assuming the ladder is leaning “tall” so that y ≥ x, the positive root corresponds to y and the negative root corresponds to x.

y = 0.5(w + √(w2 – 4c2))

x = 0.5(w – √(w2 – 4c2))

So our complete solution is:

w = –c + √(c2 + l2)

y = 0.5(w + √(w2 – 4c2))

x = 0.5(w – √(w2 – 4c2))

The initial problem had l = 4 and c = 1, which leads to the solution:

w = -1 + √(17)

y = 0.5(-1 + √(17) + √(14 – 2√17) ≈ 2.76

x = 0.5(-1 + √(17) + √(14 – 2√17) ≈ 0.36

The problem seemed to be simple, but it turned out to be quite challenging and a worthwhile exercise!