Question

surface area formative
show all calculations

a) calculate the surface area.
(diagram of a composite rectangular prism with dimensions: larger prism 10.5 m, 7.6 m, 8.4 m; smaller prism 3.5 m, 4.2 m, 3.2 m, with handwritten notes and partial calculations like \fb=(3.5×3.2)×2=22.4\, etc.)

b) the solid below consists of a cylinder... (partial text, diagram of a cylinder with 3 cm (diameter), 1.5 cm (radius), 2 cm height, partial text about cylinder fitting on top)

Explanation:

Step1: Calculate surface area of large rectangular prism

The formula for the surface area of a rectangular prism is \( SA = 2(lw + lh + wh) \). For the large prism, \( l = 10.5 \, \text{m} \), \( w = 7.6 \, \text{m} \), \( h = 8.4 \, \text{m} \).
\[ \begin{align*} SA_{\text{large}} &= 2(10.5\times7.6 + 10.5\times8.4 + 7.6\times8.4) \\ &= 2(79.8 + 88.2 + 63.84) \\ &= 2(231.84) \\ &= 463.68 \, \text{m}^2 \end{align*} \]

Step2: Calculate surface area of small rectangular prism (excluding the covered face)

For the small prism, \( l = 3.5 \, \text{m} \), \( w = 3.2 \, \text{m} \), \( h = 4.2 \, \text{m} \). The covered face (where it attaches to the large prism) has area \( 3.5\times4.2 \) (assuming the attachment is on the \( 3.5\times4.2 \) face). The surface area of a rectangular prism is \( 2(lw + lh + wh) \), but we subtract twice the covered area (once from the small prism and once from the large prism, but since we already calculated the large prism's full surface, we just adjust the small prism and add the non - covered parts). Wait, actually, when two prisms are attached, the total surface area is the sum of the surface areas of both prisms minus twice the area of the overlapping face (because the overlapping face is internal and not on the surface anymore).

First, surface area of small prism: \( SA_{\text{small}} = 2(3.5\times3.2 + 3.5\times4.2 + 3.2\times4.2) \)
\[ \begin{align*} SA_{\text{small}} &= 2(11.2 + 14.7 + 13.44) \\ &= 2(39.34) \\ &= 78.68 \, \text{m}^2 \end{align*} \]
The area of the overlapping face (the face where they are attached) is \( 3.5\times4.2=14.7 \, \text{m}^2 \). So we need to subtract \( 2\times14.7 \) from the sum of the two surface areas (because this area is counted twice in the sum of \( SA_{\text{large}} \) and \( SA_{\text{small}} \), but it's not on the surface).

Step3: Calculate total surface area

Total surface area \( SA_{\text{total}}=SA_{\text{large}}+SA_{\text{small}} - 2\times14.7 \)
\[ \begin{align*} SA_{\text{total}}&=463.68 + 78.68-29.4\\ &=463.68 + 49.28\\ &=512.96 \, \text{m}^2 \end{align*} \]
Wait, maybe my initial assumption about the overlapping face was wrong. Let's re - examine the diagram. The small prism has dimensions \( 3.5 \, \text{m} \), \( 3.2 \, \text{m} \), \( 4.2 \, \text{m} \). Let's calculate the surface area of the small prism by calculating each face:

Front/Back: \( 2\times(3.5\times3.2)=22.4 \, \text{m}^2 \)

Left/Right: \( 2\times(3.2\times4.2)=26.88 \, \text{m}^2 \)

Top/Bottom: We exclude the part that is covered. Wait, the problem says "Don't do RL" (maybe right - left?) and there are some markings. Alternatively, maybe the correct way is:

The large prism: length \( L = 10.5 \), width \( W = 7.6 \), height \( H = 8.4 \)

Small prism: length \( l = 3.5 \), width \( w = 3.2 \), height \( h = 4.2 \)

The total surface area is calculated as follows:

- For the large prism: all faces.

- For the small prism: all faces except the one that is glued to the large prism. The glued face has area \( l\times h=3.5\times4.2 = 14.7 \)

So surface area of small prism to add: \( 2(lw + wh)+lw \) (wait, no). Let's use the face - by - face method:

Large prism faces:

- Front/Back: \( 2\times(L\times H)=2\times10.5\times8.4 = 176.4 \)

- Left/Right: \( 2\times(W\times H)=2\times7.6\times8.4 = 127.68 \)

- Top/Bottom: \( 2\times(L\times W)=2\times10.5\times7.6 = 159.6 \)

Small prism faces:

- Front/Back: \( 2\times(l\times w)=2\times3.5\times3.2 = 22.4 \)

- Left/Right: \( 2\times(w\times h)=2\times3.2\times4.2 = 26.88 \)

- Top: \( l\times w = 3.5\times3.2 = 11.2 \) (assuming bottom is covered? No, the diagram is a bit unclear, but from the initial calculations in the image, FB (front - back) for small was \( (3.5\times3.2)\times2 = 22.4 \), another FB \( (3.5\times4.2)\times2 \) (maybe left - right?), and RL \( (3.2\times4.2)\times2 \). But the red marks show some corrections.

Alternatively, using the initial wrong - but - following - the - image - like approach:

From the image, for small prism:

FB (front - back) = \( (3.5\times3.2)\times2=22.4 \)

Another FB (maybe left - right) = \( (3.5\times4.2)\times2 = 29.4 \) (but it's crossed out, maybe we don't use it)

RL (right - left) = \( (3.2\times4.2)\times2 = 26.88 \) (also crossed out)

Wait, the problem says "Don't do RL" and there are covered areas. Let's start over with the correct method for composite solids:

Total surface area = Surface area of large rectangular prism + Surface area of small rectangular prism - 2×Area of the overlapping face.

Surface area of large rectangular prism: \( S_1=2(l_1w_1 + l_1h_1+w_1h_1)=2(10.5\times7.6 + 10.5\times8.4+7.6\times8.4)=2(79.8 + 88.2+63.84)=2\times231.84 = 463.68 \, \text{m}^2 \)

Surface area of small rectangular prism: \( S_2=2(l_2w_2 + l_2h_2+w_2h_2)=2(3.5\times3.2+3.5\times4.2 + 3.2\times4.2)=2(11.2 + 14.7+13.44)=2\times39.34 = 78.68 \, \text{m}^2 \)

Area of overlapping face (the face where they are joined): Let's assume the overlapping face is \( l_2\times h_2=3.5\times4.2 = 14.7 \, \text{m}^2 \)

Total surface area \( S = S_1+S_2-2\times14.7=463.68 + 78.68-29.4 = 512.96 \, \text{m}^2 \)

Answer:

The total surface area of the composite solid is \( \boldsymbol{512.96 \, \text{square meters}} \) (assuming the overlapping face area is \( 3.5\times4.2 \), the answer may vary slightly depending on the exact overlapping face, but this is the general method for calculating the surface area of composite rectangular prisms).