To find the total surface area of the toy, which consists of a cone on top of a hemisphere with a common radius of 4 cm and a cone height of 10 cm, follow the steps below:
### Step 1: Calculate the Surface Area of the Hemisphere
A hemisphere is half of a sphere. The formula for the surface area of a sphere is [tex]\(4\pi r^2\)[/tex], so the surface area of a hemisphere is half of that:
[tex]\[ \text{Surface Area of Hemisphere} = 2\pi r^2 \][/tex]
With [tex]\( r = 4 \)[/tex] cm:
[tex]\[ \text{Surface Area of Hemisphere} = 2\pi (4)^2 = 2\pi \times 16 = 32\pi \][/tex]
### Step 2: Calculate the Slant Height of the Cone
The slant height [tex]\( l \)[/tex] of a cone can be found using the Pythagorean theorem. Given the radius [tex]\( r = 4 \)[/tex] cm and the vertical height [tex]\( h = 10 \)[/tex] cm:
[tex]\[ l = \sqrt{r^2 + h^2} = \sqrt{4^2 + 10^2} = \sqrt{16 + 100} = \sqrt{116} = 2\sqrt{29} \][/tex]
### Step 3: Calculate the Surface Area of the Cone
The surface area of the cone (excluding the base) is given by the formula:
[tex]\[ \text{Surface Area of Cone} = \pi r l \][/tex]
Using [tex]\( r = 4 \)[/tex] cm and [tex]\( l = 2\sqrt{29} \)[/tex]:
[tex]\[ \text{Surface Area of Cone} = \pi \times 4 \times 2\sqrt{29} = 8\pi\sqrt{29} \][/tex]
### Step 4: Calculate the Total Surface Area of the Toy
The total surface area is the sum of the surface areas of the hemisphere and the cone:
1. Surface area of the hemisphere is [tex]\(32\pi \)[/tex].
2. Surface area of the cone is [tex]\( 8\pi\sqrt{29} \)[/tex].
Total Surface Area:
[tex]\[ \text{Total Surface Area} = 32\pi + 8\pi\sqrt{29} \][/tex]
Thus, the total surface area of the toy is:
[tex]\[ \boxed{32\pi + 8\pi\sqrt{29}} \][/tex]
