Consider two congruent triangular prisms. Each rectangular face of prism A has a width of x + 2 and each rectangular face of prism B has a length of 2x + 4. If each rectangular face of prism A has an area of 10x + 20, what is the volume of prism B? (round to nearest whole number in cm3)

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

108 cm3

Step-by-step explanation:

l x w = A

(x + 2) (2x + 4) = 10x + 20

x = 3

Then, use the width expression to find the side lengths of the equilateral triangle.

x + 2

3 + 2

5

Then, find the area of the equilateral triangle.

A =

3

4

S2

A =

3

4

52

A = 10.82

Then, use the length expression to find the length and multiply times the area of the triangle.

2x + 4

2 (3) + 4

6 + 4

10

Thus, V = 10 x 10.82 = 108.2