Question

Find the surface area of the composite figure

Answer 1

`\displaystyle SA_(Total) = (279 \pi)/(4) + 339 \ mm^2`

General Formulas and Concepts:

Pre-Algebra

Order of Operations: BPEMDAS

1. Brackets
2. Parenthesis
3. Exponents
4. Multiplication
5. Division
6. Addition
7. Subtraction

• Left to Right

Algebra I

• Factoring

Geometry

Shapes

Congruency

• Congruent sides and lengths

Radius Formula:
`\displaystyle r = (d)/(2)`

• d is diameter

Surface Area of a Rectangular Prism Formula: SA = 2(wl + hl + hw)

• w is width
• l is length
• h is height

Surface Area of a Cylinder Formula: SA = 2πrh + 2πr²

• r is radius
• h is height

Explanation:

Step 1: Define

Identify

[Rectangular Prism] w = 9 mm

[Rectangular Prism] l = 11 mm

[Rectangular Prism] h = 6 mm

[Cylinder] d = 9 mm

[Cylinder] h = 11 mm

Step 2: Derive

Modify Surface Area equations and combine

1. [Surface Area of a Cylinder Formula] Factor:
   `\displaystyle SA = 2(\pi rh + \pi r^2)`
2. [Surface Area of a Cylinder Formula] Divide by 2 [Semi-Cylinder]:
   `\displaystyle SA = \pi rh + \pi r^2`
3. [Surface Area of a Semi-Cylinder] Substitute in r [Radius Formula]:
   `\displaystyle SA = \pi ((d)/(2))h + \pi ((d)/(2))^2`
4. [Surface Area of a Semi-Cylinder] Evaluate exponents:
   `\displaystyle SA = \pi ((d)/(2))h + \pi ((d^2)/(4))`
5. [Surface Area of a Semi-Cylinder] Multiply:
   `\displaystyle SA = (\pi dh)/(2) + (\pi d^2)/(4)`
6. [Surface Area of a Rectangular Prism] Remove top:
   `\displaystyle SA = 2(wh + lh) + lw`
7. Combine Surface Area equations:
   `\displaystyle SA_(Total) = (\pi dh)/(2) + (\pi d^2)/(4) + 2(wh + lh) + lw`

Step 3: Find Surface Area

1. Substitute in variables [Combined Surface Area equation]:
   `\displaystyle SA_(Total) = (\pi (9 \ mm)(11 \ mm))/(2) + (\pi (9 \ mm)^2)/(4) + 2[(9 \ mm)(6 \ mm) + (11 \ mm)(6 \ mm)] + (11 \ mm)(9 \ mm)`
2. Evaluate exponents:
   `\displaystyle SA_(Total) = (\pi (9 \ mm)(11 \ mm))/(2) + (\pi (81 \ mm^2))/(4) + 2[(9 \ mm)(6 \ mm) + (11 \ mm)(6 \ mm)] + (11 \ mm)(9 \ mm)`
3. Multiply:
   `\displaystyle SA_(Total) = (99\pi \ mm^2)/(2) + (81\pi \ mm^2)/(4) + 2[54 \ mm^2 + 66 \ mm^2] + 99 \ mm^2`
4. [Brackets] Add:
   `\displaystyle SA_(Total) = (99\pi \ mm^2)/(2) + (81\pi \ mm^2)/(4) + 2[120 \ mm^2] + 99 \ mm^2`
5. Multiply:
   `\displaystyle SA_(Total) = (99\pi \ mm^2)/(2) + (81\pi \ mm^2)/(4) + 240 \ mm^2 + 99 \ mm^2`
6. Add:
   `\displaystyle SA_(Total) = (279 \pi)/(4) + 339 \ mm^2`

Answer 2

solution given:

For Cuboid

length[l]=11mm

breadth [b]=9mm

height[h]=6mm

For semi cylinder

height[H]=11mm

radius[r]=
`(9)/(2)=4.5mm`

Now

Totalsurface area=2(lb+bh+lh)+½(2πr(r+H))-l*b[/tex]

:2(11*9+9*6+11*6)+22/7*4.5(4.5+11)-11*9

:438+219.2-99

:558.2mm²

Here area of base is subtracted as it is not included.

Total surface area of composite figure is :558.2mm².