The volume of prism B is 108 cm³
Step-by-step explanation:
* Lets study the information to solve the problem
- Any triangular prism has five faces, two of them are triangles and the
other three are rectangles
- Its two bases are triangles
- Its side faces are rectangles
- The volume of it is its base area × its height
- The two triangular prisms are congruent, then all corresponding
dimensions are equal and their surface areas and volumes are equal
* Now lets solve the problem
∵ The two triangular prisms are congruent
∴ All corresponding faces are congruent
∵ The width of each rectangular faces in prism A = x + 2
∴ The width of each rectangular faces in prism B = x + 2
- The side of the triangular base is the width of the rectangular face
∴ All sides of the triangular base in the prism B = x + 2
∵ The area of the all rectangular face in prism A = 10x + 20
∴ The area of the all rectangular face in prism B = 10x + 20
∵ The length of each rectangular face in prism B is 2x + 2
- The length of the rectangular face of the triangular prism is its height
∴ The height of the prism b = 2x + 4
* Now lets find the value of x
∵ The rectangular face of prism B has width x + 2 , length 2x + 4
and area 10x + 20
∵ The area of the rectangle = length × width
∴ (2x + 4) × (x + 2) = 10x + 20 ⇒ simplify by using foil method
∵ 2x(x) + 2x(2) + 4(x) + 4(2) = 10x + 20
∴ 2x² + 4x + 4x + 8 = 10x + 20 ⇒ add the like term
∴ 2x² + 8x + 8 = 10x + 20 ⇒ subtract 10 x from both sides
∴ 2x² - 2x + 8 = 20 ⇒ subtract 20 from both sides
∴ 2x² - 2x - 12 = 0 ⇒ divide all terms by 2 to simplify
∴ x² - x - 6 = 0 ⇒ factorize it into two factors
∵ x² = x × x
∵ -6 = -3 × 2
∵ -3x + 2x = -x
∴ (x - 3)(x + 2) = 0
- Equate each bracket by 0
∴ x - 3 = 0 ⇒ add 3 to both sides
∴ x = 3
OR
∴ x + 2 = 0 ⇒ subtract 2 from both sides
∴ x = -2 ⇒ we will refuse this value of x because there is no
negative dimensions
∴ The value of x is 3 only
- Lets find the dimensions of the prism B
∵ Its width = x + 2
∴ Its width = 3 + 2 = 5 cm
∴ The sides of the triangular base are 5 cm
∵ The triangular base is equilateral triangle
∵ The area of any equilateral triangle = √3/4 (side)²
∴ The area of the base = (√3/4) (5)² = 25√3/4 cm²
∵ The height of the prism B = 2x + 4
∵ x = 3
∴ The height = 2(3) + 4 = 6 + 4 = 10 cm
∵ The volume of any prism = its base area × its height
∴ The volume of prism B = 25√3/4 × 10 ≅ 108 cm³
* The volume of prism B is 108 cm³
