A right pyramid,with a square base and four equal isosceles triangles ,is cut from a square piece of paper which is 3x cm long.If the area of the shaded part is 54 cm2, find the value of x.Please help me to solve this problem.

To solve this problem, let's break it down step by step:

Step 1: Understand the problem:
We have a square piece of paper with side length 3x cm. We need to find the value of x. The paper is cut to form a right pyramid with a square base and four equal isosceles triangles. We are given that the area of the shaded part is 54 cm².

Step 2: Visualize the problem:
Visualize a square piece of paper. It is cut and folded to form a right pyramid with a square base and four isoceles triangles. One of the triangles is shaded.

Step 3: Find the total area of the pyramid:
The total surface area of the pyramid can be calculated by adding the areas of the square base and the four triangles. Since the base is a square, its area is side length squared. So, the area of the base = (3x)² = 9x² cm².

Each of the four triangles is isosceles, which means it has two equal sides. Let's say the equal side length is y cm. The height of each triangle is also y cm.

The area of each triangle can be calculated using the formula: 0.5 * base * height. In this case, the base is y cm and the height is y cm, so the area of each triangle is 0.5 * y * y = 0.5y² cm².

Since there are four equal triangles, the total area of the four triangles can be calculated as 4 times the area of one triangle, so the total area of the four triangles = 4 * 0.5y² = 2y² cm².

Thus, the total area of the pyramid = area of the base + area of the four triangles = 9x² + 2y² cm².

Step 4: Find the area of the shaded part:
We are given that the area of the shaded part is 54 cm². So, we can equate the total area of the pyramid to 54:

9x² + 2y² = 54

Step 5: Solve for x:
To find the value of x, we need another equation. Notice that each side of the pyramid is 3x cm long, and each base side is y cm long. Since the pyramid is a right pyramid, the sides and the base are connected by a right angle. Therefore, we can apply the Pythagorean theorem to find the relationship between x and y.

From the Pythagorean theorem, we know that the sum of the squares of the two legs of a right triangle is equal to the square of the hypotenuse.

In this case, the two legs are x and x, and the hypotenuse is y. So, we can write the equation:

x² + x² = y²
2x² = y²

Step 6: Substitute the value of y² into the equation for the area of the pyramid:
Substitute 2x² for y² in the equation 9x² + 2y² = 54:

9x² + 2(2x²) = 54
9x² + 4x² = 54
13x² = 54

Step 7: Solve for x:

Divide both sides by 13:

x² = 54 / 13
x² ≈ 4.154

Take the square root of both sides to solve for x:

x ≈ √(4.154)
x ≈ 2.04

Therefore, the value of x is approximately 2.04 cm.