example find the surface area of the pyramid. use the formula: $sa = \frac{1}{2}ps + b$. step 1 find the facts you need to use the formula. the slant height s is 4.3 feet. find p, the perimeter of the base: 3 + 3 + 3 + 3 = 12. find b, the area of the base: $3^{2}=9$. step 2 evaluate the formula. $sa=\frac{1}{2}ps + b=\frac{1}{2}cdot12cdot4.3 + 9=34.8$ sq ft. the surface area is 34.8 sq ft. find the surface area of each figure. if necessary, round your answer to the nearest tenth square unit. 1. 4 m 8 m 5 m 2. 10 yd 3. 3 yd 9 yd 4. 2 cm 2 cm 2 cm 5. 6 ft 6 ft 5.8 ft 6. 2 m 4.5 m. solve as directed. 7. the illustration shows a truck bed with no top. what is the surface area of the inside of the truck bed? 11 ft 3 ft 8 ft 8. crunchy oats cereal comes in a box that measures 8 by 11 by 2 inches. bran crisps cereal comes in a box that measures 7 by 10 by 3 inches. (hint: make a drawing with labels.) a. which box has a larger surface area? b. which box has a greater volume?

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1. **For the rectangular - prism in problem 1 (dimensions $4m\times8m\times5m$)**:
   - The surface - area formula for a rectangular prism is $SA = 2(lw+lh + wh)$, where $l = 4m$, $w = 8m$, and $h = 5m$.
   - **Step 1: Calculate $lw$, $lh$, and $wh$**
     - $lw=4\times8 = 32m^{2}$
     - $lh = 4\times5=20m^{2}$
     - $wh = 8\times5 = 40m^{2}$
   - **Step 2: Calculate the surface area**
     - $SA=2(32 + 20+40)$
     - First, add the values inside the parentheses: $32 + 20+40=92m^{2}$
     - Then, multiply by 2: $SA = 2\times92=184m^{2}$
2. **For the sphere in problem 2 (radius $r = 10yd$)**:
   - The surface - area formula for a sphere is $SA = 4\pi r^{2}$.
   - **Step 1: Square the radius**
     - $r^{2}=10^{2}=100yd^{2}$
   - **Step 2: Calculate the surface area**
     - $SA = 4\pi\times100$
     - $SA = 400\pi\approx400\times3.14 = 1256.0yd^{2}$
3. **For the cylinder in problem 3 (radius $r = 3yd$, height $h = 9yd$)**:
   - The surface - area formula for a cylinder is $SA=2\pi r^{2}+2\pi rh$.
   - **Step 1: Calculate $2\pi r^{2}$**
     - $r^{2}=3^{2}=9yd^{2}$, so $2\pi r^{2}=2\pi\times9 = 18\pi yd^{2}\approx18\times3.14 = 56.52yd^{2}$
   - **Step 2: Calculate $2\pi rh$**
     - $2\pi rh=2\times3.14\times3\times9=169.56yd^{2}$
   - **Step 3: Calculate the surface area**
     - $SA=18\pi + 169.56$
     - $SA\approx56.52+169.56 = 226.1yd^{2}$
4. **For the cube in problem 4 (side length $s = 2cm$)**:
   - The surface - area formula for a cube is $SA = 6s^{2}$.
   - **Step 1: Square the side length**
     - $s^{2}=2^{2}=4cm^{2}$
   - **Step 2: Calculate the surface area**
     - $SA = 6\times4=24cm^{2}$
5. **For the square - based pyramid in problem 5 (side length of base $s = 6ft$, slant height $l = 5.8ft$)**:
   - The surface - area formula is $SA=\frac{1}{2}ps + B$, where $p$ is the perimeter of the base and $B$ is the area of the base.
   - **Step 1: Calculate $p$ and $B$**
     - $p = 4\times6=24ft$
     - $B = 6^{2}=36ft^{2}$
   - **Step 2: Calculate the surface area**
     - $SA=\frac{1}{2}\times24\times5.8+36$
     - First, calculate $\frac{1}{2}\times24\times5.8=12\times5.8 = 69.6ft^{2}$
     - Then, add the area of the base: $SA=69.6 + 36=105.6ft^{2}$
6. **For the cone in problem 6 (radius $r = 2m$, slant height $l = 4.5m$)**:
   - The surface - area formula for a cone is $SA=\pi r^{2}+\pi rl$.
   - **Step 1: Calculate $\pi r^{2}$**
     - $r^{2}=2^{2}=4m^{2}$, so $\pi r^{2}=4\pi m^{2}\approx4\times3.14 = 12.56m^{2}$
   - **Step 2: Calculate $\pi rl$**
     - $\pi rl=3.14\times2\times4.5 = 28.26m^{2}$
   - **Step 3: Calculate the surface area**
     - $SA=4\pi+28.26$
     - $SA\approx12.56 + 28.26=40.8m^{2}$
7. **For the truck - bed in problem 7 (dimensions $8ft\times3ft\times11ft$, no top)**:
   - The surface area of the inside of the truck - bed (open - top rectangular prism) is $SA=lw+2lh + 2wh$.
   - **Step 1: Calculate $lw$, $lh$, and $wh$**
     - $lw=8\times3 = 24ft^{2}$
     - $lh=8\times11 = 88ft^{2}$
     - $wh = 3\times11=33ft^{2}$
   - **Step 2: Calculate the surface area**
     - $SA=24+2\times88+2\times33$
     - First, calculate $2\times88 = 176ft^{2}$ and $2\times33 = 66ft^{2}$
     - Then, add: $SA=24 + 176+66=266ft^{2}$
8. **For the cereal - box problem in problem 8**:
   - **For Crunchy Oats (dimensions $l_1 = 8in$, $w_1 = 11in$, $h_1 = 2in$)**:
     - Surface area $SA_1=2(l_1w_1+l_1h_1+w_1h_1)$
     - $l_1w_1=8\times11 = 88in^{2}$, $l_1h_1=8\times2 = 16in^{2}$, $w_1h_1=11\times2 = 22in^{2}$
     - $SA_1=2(88 + 16+22)=2\times126 = 252in^{2}$
     - Volume $V_1=l_1w_1h_1=8\times11\times2=176in^{3}$
   - **For Bran Crisps (dimensions $l_2 = 7in$, $w_2 = 10in$, $h_2 = 3in$)**:
     - Surface area $SA_2=2(l_2w_2+l_2h_2+w_2h_2)$
     - $l_2w_2=7\times10 = 70in^{2}$, $l_2h_2=7\times3 = 21in^{2}$, $w_2h_2=10\times3 = 30in^{2}$
     - $SA_2=2(70 + 21+30)=2\times121 = 242in^{2}$
     - Volume $V_2=l_2w_2h_2=7\times10\times3=210in^{3}$
   - **a. Which box has a larger surface area?**
     - Crunchy Oats has a larger surface area ($252in^{2}>242in^{2}$)
   - **b. Which box has a greater volume?**
     - Bran Crisps has a greater volume ($210in^{3}>176in^{3}$)

# Answer:
1. $184m^{2}$
2. $1256.0yd^{2}$
3. $226.1yd^{2}$
4. $24cm^{2}$
5. $105.6ft^{2}$
6. $40.8m^{2}$
7. $266ft^{2}$
8. a. Crunchy Oats has a larger surface area.
   b. Bran Crisps has a greater volume.

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