5. this object is composed of a cylinder of diameter 4 cm and height 14 cm on top of another cylinder of…

Explanation:

Step1: Recall the formula for the surface area of a cylinder

The surface area of a cylinder is given by ( SA = 2\pi r^2 + 2\pi rh ), where ( r ) is the radius and ( h ) is the height. However, for the composite object, we need to consider the areas that are covered (the area where the smaller cylinder is attached to the larger one) and adjust accordingly.

Step2: Calculate the dimensions (radii)

For the larger cylinder: diameter ( d_1 = 12 ) cm, so radius ( r_1=\frac{12}{2}=6 ) cm, height ( h_1 = 4 ) cm. For the smaller cylinder: diameter ( d_2 = 4 ) cm, so radius ( r_2=\frac{4}{2}=2 ) cm, height ( h_2 = 14 ) cm.

Step3: Surface area of the larger cylinder (adjusted)

The surface area of the larger cylinder will have its top area reduced by the area of the base of the smaller cylinder.

• Lateral (curved) surface area of larger cylinder: ( 2\pi r_1h_1=2\pi\times6\times4 = 48\pi )
• Area of the bottom base of larger cylinder: ( \pi r_1^2=\pi\times6^2 = 36\pi )
• Area of the top base of larger cylinder (adjusted): ( \pi r_1^2-\pi r_2^2=\pi(6^2 - 2^2)=\pi(36 - 4)=32\pi ) So total surface area of larger cylinder part: ( 48\pi+36\pi + 32\pi=116\pi )

Step4: Surface area of the smaller cylinder

The smaller cylinder has no bottom base (since it's attached to the larger cylinder), so we calculate its lateral surface area and its top base area.

• Lateral surface area of smaller cylinder: ( 2\pi r_2h_2=2\pi\times2\times14 = 56\pi )
• Area of the top base of smaller cylinder: ( \pi r_2^2=\pi\times2^2 = 4\pi ) So total surface area of smaller cylinder part: ( 56\pi+4\pi = 60\pi )

Step5: Total surface area of the composite object

Add the surface areas of the adjusted larger cylinder and the smaller cylinder: ( SA_{total}=116\pi+60\pi=176\pi ) Wait, that can't be right. Wait, let's re - calculate the surface area of the larger cylinder correctly.

Alternative approach:

• Surface area of larger cylinder (full surface area) : ( 2\pi r_1^2+2\pi r_1h_1=2\pi\times6^2 + 2\pi\times6\times4=72\pi + 48\pi = 120\pi )
• Subtract the area of the circle where the smaller cylinder is attached: ( \pi r_2^2=\pi\times2^2 = 4\pi )
• Surface area of smaller cylinder (lateral surface area + top area, since bottom is attached): ( 2\pi r_2h_2+\pi r_2^2=2\pi\times2\times14+\pi\times2^2 = 56\pi+4\pi = 60\pi )

Now, total surface area: ( (120\pi - 4\pi)+60\pi=116\pi + 60\pi = 176\pi \approx176\times3.1416\approx552.92 \approx 553 )

Wait, let's check again:

Larger cylinder:

• Bottom: ( \pi r_1^2=\pi\times6^2 = 36\pi )
• Lateral: ( 2\pi r_1h_1=2\pi\times6\times4 = 48\pi )
• Top (adjusted): ( \pi r_1^2-\pi r_2^2=\pi(36 - 4)=32\pi ) Sum for larger: ( 36\pi+48\pi + 32\pi=116\pi )

Smaller cylinder:

• Lateral: ( 2\pi r_2h_2=2\pi\times2\times14 = 56\pi )
• Top: ( \pi r_2^2=\pi\times2^2 = 4\pi ) Sum for smaller: ( 56\pi+4\pi = 60\pi )

Total: ( 116\pi+60\pi = 176\pi\approx176\times3.14159265\approx552.919\approx553 )

Answer:

d. ( 553\space cm^2 )