i need help with this can i please get it fast?

Prism on top:
The top face (rectangle) is 6 x 20 = 120
The bottom face (also rectangle) is 10 x 20 = 200
The face on the left (yet another rectangle) is 5 x 20 = 100
The face on the right can be found with the pythagorean theorem. You can split the trapezoidal face in front into a 6x5 rectangle and a 4x5 triangle:
a^2 + b^2 = c^2 (a = 4, b = 5)
4^2 + 5^2 = c^2
16 + 25 = c^2
c^2 = 41
c = sqrt(41)
The face on the right is 20 x sqrt(41) which is 20sqrt(41)
Face in front is a trapezoid. There are two trapezoids, front and back. Area ia (a+b) x 1/2 x h, where a is 6 and b = 10. h = 5.
(6 + 10) x 1/2 x 5 = 16 x 1/2 x 5 = 8 x 5 = 40
Since there are two faces, 40 x 2 = 80.
Adding them up:
120 + 200 + 100 + 80 = 500 cm^2

sqdancefan sqdancefan · Answer 2 · 2024-03-30T18:31:09+01:00

Answer:

trapezoidal prism: 628.06 cm²
triangular prism: 678.43 cm²

Step-by-step explanation:

You want the surface area of the prisms shown.

Surface area

The area of each prism will be the sum of the areas of the two bases and the areas of the rectangular lateral faces. The lateral area is the product of the prism height and the perimeter of the base.

In each case here, some work must be done to find the length of the unmarked side of the base. That can be accomplished using the Pythagorean theorem one or more times.

Trapezoidal prism

The length of the unknown side is the hypotenuse of the right triangle with height 5 cm and base (10 cm -6 cm) = 4 cm. Using the Pythagorean theorem, we find that length to be ...

x = √(5² +4²) = √41

The area of the two bases is ...

A = 2(1/2)(b1 +b2)h = (b1 +b2)h

A = (6 cm +10 cm)(5 cm) = 80 cm²

The lateral area is ...

A = Ph = (6 +5 +10 +√41 cm)(20 cm) = 20(21 +√41) cm²

The total area is ...

SA = (80 cm²) +20(21 +√41) cm² = 20(25 +41) cm²

SA ≈ 628.06 cm²

Triangular prism

The length of the unknown side of the triangle can be found using the Pythagorean theorem twice. The first application will find the length of the base of the triangle to the left of the altitude.

x = √(11² -9²) = √40 = 2√10

The length of the base to the right of the altitude is then ...

y = 14 -x = 14 -2√10

The unknown side is the hypotenuse of the right triangle with height 9 cm and base (14 -2√10) cm. Its length will be ...

z = √(9² +(14 -2√10)²) = √(81 +(196 +40 -56√10)) = √(317 -56√10)

The base area of the prism is ...

A = 2(1/2)bh = bh

A = (14 cm)(9 cm) = 126 cm²

The lateral area is ...

A = Ph = (11 +14 +√(317-56√10))·15 cm² ≈ 552.43 cm²

The total area is ...

SA = 126 cm² +552.43 cm²

SA = 678.43 cm²