Question

a firework rocket consists of a cone stacked on top of a cylinder, where the radii of the cone and the cylinder are equal. the diameter of the cylindrical base of the rocket is 8 in and the height of the cylinder is 5 in, while the height of the cone is 3 in. calculate the surface area of the rocket. leave your answer in terms of π. 184π sq. in. 76π sq. in. 168π sq. in. 98π sq. in.

Explanation:

Step1: Find the radius

The diameter of the base is 8 in, so the radius $r=\frac{8}{2}=4$ in.

Step2: Calculate the lateral - surface area of the cylinder

The formula for the lateral - surface area of a cylinder is $A_{cylinder - lateral}=2\pi rh$. Here, $r = 4$ in and $h = 5$ in, so $A_{cylinder - lateral}=2\pi\times4\times5=40\pi$ sq in.

Step3: Calculate the lateral - surface area of the cone

The formula for the slant height of a cone $l=\sqrt{r^{2}+h^{2}}$, where $r = 4$ in and $h = 3$ in. So $l=\sqrt{4^{2}+3^{2}}=\sqrt{16 + 9}=\sqrt{25}=5$ in. The formula for the lateral - surface area of a cone is $A_{cone - lateral}=\pi rl$. Substituting $r = 4$ in and $l = 5$ in, we get $A_{cone - lateral}=\pi\times4\times5=20\pi$ sq in.

Step4: Calculate the base area of the cylinder

The formula for the base area of a cylinder is $A_{base}=\pi r^{2}$. Substituting $r = 4$ in, we get $A_{base}=\pi\times4^{2}=16\pi$ sq in.

Step5: Calculate the total surface area

The total surface area of the rocket $A=A_{cylinder - lateral}+A_{cone - lateral}+A_{base}=40\pi+20\pi + 16\pi=76\pi$ sq in.

Answer:

$76\pi$ sq in