Prism whose base is a trapezoid.the surface area is 27cm².find the value of x?

To find the value of x, we need to first calculate the area of the trapezoid base of the prism.

Let the trapezoid base have bases of length a and b, and height h. The area of a trapezoid is given by the formula:

Area = 1/2 * (a + b) * h

Given that the surface area of the prism is 27 cm², and the area of the trapezoid base is 1/2 * (a + b) * h, we can set up the equation:

27 = 2 * (1/2 * (a + b) * h) + 2 * (a + b) * x

27 = (a + b) * h + 2 * (a + b) * x

27 = (a + b) * (h + 2x)

Since the surface area is given as 27 cm², we know that the surface area is equal to the sum of the areas of the four sides of the prism. The area of each side of the prism is given by the formula:

Area = base * height

Since the base of the prism is a trapezoid, the area of each side is given by:

Area = (a + b) * x

Therefore, the surface area of the prism is given by:

27 = 2 * (a + b) * x

27 = 2 * (a + b) * x

27 = (a + b) * x

Now we have two equations:

27 = (a + b) * (h + 2x)

27 = (a + b) * x

We can solve these two equations simultaneously to find the value of x.

Given:
- Prism with a trapezoid base
- Surface area = 27 cm²

To find:
- Value of x

Steps:

1. Identify the trapezoid's dimensions:

Let the bases of the trapezoid be a and b, and the height be h. From the surface area formula, we have:

Surface area = 2(area of bases) + area of lateral surface

2. Calculate the area of the trapezoid bases:

Area of trapezoid = (a + b)h / 2

For the lower base: a = 2x
For the upper base: b = 3x

3. Calculate the area of the lateral surface:

Lateral surface area = perimeter of base x height

Perimeter of base = a + b + 2h = 2x + 3x + 2h = 5x + 2h

Lateral surface area = (5x + 2h)h

4. Substitute and simplify:

Surface area = 2(area of bases) + area of lateral surface

27 = 2[(2x + 3x)h / 2] + (5x + 2h)h

27 = 2(5xh / 2) + 5xh² + 2h²

27 = 5xh + 5xh² + 2h²

5. Rearrange and solve for h:

5xh² + 7xh - 27 = 0

Using the quadratic formula:

h = (-7 ± √(7² - 4(5)(-27))) / (2(5))

h = (-7 ± √361) / 10

h = (-7 ± 19) / 10

h = 1.2 or -3.6 (discarding the negative solution)

6. Substitute h back into the equation for the lateral surface area:

Lateral surface area = (5x + 2h)h

Lateral surface area = (5x + 2(1.2))1.2

Lateral surface area = 7.2x

7. Subtract the lateral surface area from the total surface area to get the sum of the areas of the bases:

27 - 7.2x = 5xh

8. Solve for x:

5xh = 19.8

x = 3.96 (rounded to two decimal places)

Therefore, the value of x is approximately 3.96.