Question

section 1.4 - surface area and volume of prisms • 27
1.
i)

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v =

ii)

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v =

iii)

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v =

iv)

s.a. =
v =

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Answer:

Part 1 (a) - Rectangular Prism (8m, 12m, 4m, 3m, 5m? Wait, maybe dimensions: length=12m, width=8m, height=4m? Wait, the diagram: 8m, 12m, 4m, 3m, 5m? Wait, maybe it's a rectangular prism with length \( l = 12 \) m, width \( w = 8 \) m, height \( h = 4 \) m? Wait, no, maybe the base is a rectangle with 8m and 12m, and height 4m? Wait, surface area of a rectangular prism is \( 2(lw + lh + wh) \), volume is \( lwh \). Wait, maybe the dimensions are length=12, width=8, height=4? Wait, let's check:

Step 1: Surface Area of Rectangular Prism

The formula for the surface area (SA) of a rectangular prism is \( SA = 2(lw + lh + wh) \), where \( l \) is length, \( w \) is width, \( h \) is height.
Given \( l = 12 \) m, \( w = 8 \) m, \( h = 4 \) m (assuming from the diagram).

$$ SA = 2(12 \times 8 + 12 \times 4 + 8 \times 4) $$

$$ = 2(96 + 48 + 32) = 2(176) = 352 \, \text{m}^2 $$

Step 2: Volume of Rectangular Prism

The formula for the volume (V) of a rectangular prism is \( V = lwh \).

$$ V = 12 \times 8 \times 4 = 384 \, \text{m}^3 $$

Part 1 (b) - Composite Prism (9m, 7m, 8m, and a smaller part? Wait, the diagram: a larger rectangular prism and a smaller one? Wait, maybe the composite figure: larger prism: length=7m, width=8m, height=9m? No, maybe the total length is 7m + 7m? Wait, no, the diagram shows a 9m height, 8m width, and the base is 7m and another 7m? Wait, maybe it's a composite of two rectangular prisms: one with dimensions 9m (height), 8m (width), 7m (length), and another with dimensions (9 - 4)? Wait, no, maybe the correct approach is to calculate the surface area by considering the composite. Alternatively, maybe the dimensions are: total length=7m, height=9m, width=8m, and a smaller prism attached? Wait, maybe I misread. Alternatively, let's assume it's a rectangular prism with length=7m, width=8m, height=9m, but with a notch? No, maybe the surface area is calculated as the surface area of the larger prism minus the area of the two notched faces. Wait, this is getting complicated. Maybe the problem is simpler: perhaps the composite figure has length=7m, width=8m, height=9m, and the smaller part is 4m? Wait, maybe the correct dimensions are: length=7m, width=8m, height=9m, and the other part is 7m (so total length 14m? No, the diagram shows 7m and 7m? Wait, maybe the original figure is a rectangular prism with length=14m, width=8m, height=9m, but with a smaller prism (length=7m, width=8m, height=4m) removed? No, this is unclear. Alternatively, maybe the problem is a rectangular prism with length=7m, width=8m, height=9m, and another prism with length=7m, width=8m, height=4m? No, perhaps I should skip this part for now and move to the triangular prism.

Part 2 (c) - Triangular Prism (3mm, 4mm, 5mm, 2mm)

A triangular prism has two triangular bases and three rectangular faces.

• Triangular base: legs 3mm, 4mm (right triangle, since \( 3^2 + 4^2 = 5^2 \)), hypotenuse 5mm.
• Length of the prism (distance between the triangular bases) is 2mm.

Step 1: Surface Area of Triangular Prism

Surface area (SA) = 2(Area of triangular base) + (Perimeter of triangular base) × length of prism.

• Area of triangular base: \( \frac{1}{2} \times 3 \times 4 = 6 \, \text{mm}^2 \)
• Perimeter of triangular base: \( 3 + 4 + 5 = 12 \, \text{mm} \)
• Length of prism: 2mm

$$ SA = 2(6) + (12 \times 2) = 12 + 24 = 36 \, \text{mm}^2 $$

Step 2: Volume of Triangular Prism

Volume (V) = Area of triangular base × length of prism.

$$ V = 6 \times 2 = 12 \, \text{mm}^3 $$

Part 2 (d) - Triangular Prism (4ft, 5ft, 6ft, 12ft)

Triangular base: base=6ft, height=4ft (right triangle), hypotenuse=5ft? Wait, \( 3^2 + 4^2 = 5^2 \), but here base=6ft, height=4ft? Wait, no, the triangular base has base=6ft, height=4ft (since it's a right triangle with height 4ft, base 6ft). The length of the prism (distance between the triangular bases) is 12ft.

Step 1: Surface Area of Triangular Prism

• Area of triangular base: \( \frac{1}{2} \times 6 \times 4 = 12 \, \text{ft}^2 \)
• Perimeter of triangular base: \( 6 + 5 + \sqrt{6^2 - 4^2} \)? Wait, no, the sides are 6ft (base), 5ft (hypotenuse), and 4ft (height)? Wait, no, the triangle has sides 6ft (base), 5ft (one leg), and 4ft (height)? Wait, no, the right triangle: if height is 4ft, base is 6ft, then hypotenuse is \( \sqrt{4^2 + 6^2} = \sqrt{52} \approx 7.21 \), but the diagram shows 5ft. Wait, maybe the triangular base is a right triangle with legs 3ft, 4ft, hypotenuse 5ft, and base 6ft? No, that doesn't match. Wait, the diagram shows 4ft (height), 6ft (base), 5ft (side), 12ft (length of prism).

Wait, correct approach:

• Triangular base: base \( b = 6 \) ft, height \( h = 4 \) ft (so area \( \frac{1}{2} \times 6 \times 4 = 12 \, \text{ft}^2 \))
• The other sides of the triangle: one is 5ft (the slant side), and the length of the prism is 12ft.

Surface area (SA) = 2(Area of triangle) + (Perimeter of triangle) × length of prism.

• Perimeter of triangle: \( 6 + 5 + \) the third side. Wait, the third side: since it's a triangle with base 6, height 4, and one side 5, let's check: if height is 4, then the base is split into two segments: let’s say \( x \) and \( 6 - x \). Then \( x^2 + 4^2 = 5^2 \) → \( x^2 = 25 - 16 = 9 \) → \( x = 3 \). So the third side is \( \sqrt{(6 - 3)^2 + 4^2} = \sqrt{9 + 16} = 5 \). So the triangle is isoceles with two sides 5ft, base 6ft, height 4ft.

Thus, perimeter of triangle: \( 6 + 5 + 5 = 16 \, \text{ft} \)

$$ SA = 2(12) + (16 \times 12) = 24 + 192 = 216 \, \text{ft}^2 $$

Step 2: Volume of Triangular Prism

Volume (V) = Area of triangular base × length of prism.

$$ V = 12 \times 12 = 144 \, \text{ft}^3 $$

Summary (for now, focusing on the triangular prism part):

Part 1 (a) (Rectangular Prism):

• SA: \( 352 \, \text{m}^2 \)
• V: \( 384 \, \text{m}^3 \)

Part 2 (c) (Triangular Prism - 3mm, 4mm, 5mm, 2mm):

• SA: \( 36 \, \text{mm}^2 \)
• V: \( 12 \, \text{mm}^3 \)

Part 2 (d) (Triangular Prism - 4ft, 6ft, 5ft, 12ft):

• SA: \( 216 \, \text{ft}^2 \)
• V: \( 144 \, \text{ft}^3 \)

(Note: For part 1 (b), the composite prism, more clarity on dimensions is needed. If you can provide the exact dimensions, I can re-calculate.)