The formula for the surface area of a half-cylinder must include one-half of the surface area of a cylinder, which would be:

\(\displaystyle SA = \frac{1}{2} (lateral\ area + 2 \cdot (base))\)

We also need to add the area of the new rectangular face that is created by cutting the cylinder in half. The area of this rectangle would be:

\(\displaystyle A=l\times w\)

where the length of the rectangle is the same as the height of the half-cylinder, and the width of the rectangle is the same as the diameter of the base of the half-cylinder. So we can rewrite the area of the rectangle as:

\(\displaystyle A=h\p\times d\)

Now we can combine the two area formulas to find the total surface area of the half-cylinder:

\(\displaystyle SA = \frac{1}{2} (2 \pi r (h) + 2 (\pi r^2))+h\cdot d\)

\(\displaystyle SA = \pi r (h) + \pi r^2+hd\)

where \(\displaystyle r\) is the radius of the base and \(\displaystyle h\) is the length of the height, and \(\displaystyle d\) is the diameter of the base.

Plugging in our values, we get:

\(\displaystyle SA = \pi (5m) (15m) + \pi (5m)^2+15\cdot 10\)

\(\displaystyle SA = 75 \pi m^2 + 25 \pi m^2+150 = 150+100 \pi m^2\)

The formula for the surface area of the polyhedron is:

\(\displaystyle SA = SA_{cone}+\frac{1}{2}SA_{sphere}\)

\(\displaystyle SA = \pi r l+\frac{1}{2}(4 \pi r^2)\)

\(\displaystyle SA = \pi r l+2 \pi r^2\)

Where \(\displaystyle r\) is the radius of the cone, \(\displaystyle l\) is the slant height of the cone, and \(\displaystyle r\) is the radius of the sphere

Use the formula for a \(\displaystyle 30-60-90\) triangle to find the radius and slant height:

\(\displaystyle a-a\sqrt{2}-2a\)

\(\displaystyle 3\sqrt{3}m-9m-6\sqrt{3}m\)

Plugging in our values, we get:

\(\displaystyle SA = \pi r l+2 \pi r^2\)

\(\displaystyle SA = (\pi)(3\sqrt{3}m)(6\sqrt{3}m)+2(\pi)(3\sqrt{3}m)^2\)

\(\displaystyle SA = 54 \pi m^2 + 54 \pi m^2 = 108 \pi m^2\)

The formula for the surface area of the polyhedron is:

\(\displaystyle SA = (cone:no\ base)+(cylinder:one\ base)\)

\(\displaystyle SA = \frac{1}{2}(circumference)(slant\ height)+(base)+(circumference)(height)\)

\(\displaystyle SA = \frac{1}{2}(2 \pi r)(h_s)+(\pi r^2)+(2 \pi r)(h)\)

where \(\displaystyle r\) is the radius of the cone, \(\displaystyle h_s\) is the slant height of the cone, \(\displaystyle r\) is the radius of the cylinder, and \(\displaystyle h\) is the height of the cylinder.

Use the formula for a \(\displaystyle 30-60-90\) triangle to find the length of the radius:

\(\displaystyle a-a\sqrt{3}-2a\)

\(\displaystyle 3m-3\sqrt{3}m-6m\)

Plugging in our values, we get:

\(\displaystyle SA = \frac{1}{2}(2\pi (3m))(6m)+(\pi)(3m)^2+(2\pi (3m)(9m))\)

\(\displaystyle SA = 81 \pi m^2\)

The formula for the surface area of a polyhedron is:

\(\displaystyle SA = (cone)+ \frac{1}{2}(sphere)\)

\(\displaystyle SA = \frac{1}{2}(2\pi r)(h_s)+ \frac{1}{2}(4\pi r^2)\)

\(\displaystyle SA = \pi rh_s+ 2\pi r^2\)

where \(\displaystyle r\) is the radius of the polyhedron and \(\displaystyle h_s\) is the slant height of the cone.

Use the formula for a \(\displaystyle 45-45-90\) triangle to find the length of the radius:

\(\displaystyle a-a-a\sqrt{2}\)

\(\displaystyle 4\sqrt{2}m-4\sqrt{2}m-8m\)

Plugging in our values, we get:

\(\displaystyle SA = \pi (4\sqrt{2}m)(8m)+ 2\pi (4\sqrt{2}m)^2\)

\(\displaystyle SA = 32\sqrt{2}\pi + 64 \pi m^2\)