To find the total surface area of the rocket, we need to calculate the area of its exposed surfaces. The…

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To find the total surface area of the rocket, we need to calculate the area of its exposed surfaces. The rocket is composed of a cylinder and a cone. The exposed surfaces are:

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To find the total surface area of the rocket, we need to calculate the area of its exposed surfaces. The rocket is composed of a cylinder and a cone. The exposed surfaces are:

The circular base of the cylinder.
The lateral surface area of the cylinder.
The lateral surface area of the cone.

The circular face where the cylinder and cone meet is internal and not part of the total external surface area.

From the image, we have the following dimensions:

Diameter of the cylinder (and base of the cone) = $4 \text{ cm}$ .
Radius $r = \frac{4 \text{ cm}}{2} = 2 \text{ cm}$ .
Height of the cylinder $h_{cyl} = 20 \text{ cm}$ .
Height of the cone $h_{cone} = 15 \text{ cm}$ .

Calculate the slant height ( $l$ ) of the cone.
The slant height is found using the Pythagorean theorem:
$l = \sqrt{r^2 + h_{cone}^2}$ $l = \sqrt{(2 \text{ cm})^2 + (15 \text{ cm})^2}$ $l = \sqrt{4 \text{ cm}^2 + 225 \text{ cm}^2}$ $l = \sqrt{229} \text{ cm}$
Calculate the area of the circular base of the cylinder.
$A_{base} = \pi r^2$ $A_{base} = \pi (2 \text{ cm})^2$ $A_{base} = 4\pi \text{ cm}^2$
Calculate the lateral surface area of the cylinder.
$A_{cyl\_lat} = 2\pi r h_{cyl}$ $A_{cyl\_lat} = 2\pi (2 \text{ cm}) (20 \text{ cm})$ $A_{cyl\_lat} = 80\pi \text{ cm}^2$
Calculate the lateral surface area of the cone.
$A_{cone\_lat} = \pi r l$ $A_{cone\_lat} = \pi (2 \text{ cm}) (\sqrt{229} \text{ cm})$ $A_{cone\_lat} = 2\pi \sqrt{229} \text{ cm}^2$
Calculate the total surface area ( $A_{total}$ ) by summing the areas of the exposed surfaces.
$A_{total} = A_{base} + A_{cyl\_lat} + A_{cone\_lat}$ $A_{total} = 4\pi \text{ cm}^2 + 80\pi \text{ cm}^2 + 2\pi \sqrt{229} \text{ cm}^2$ $A_{total} = (84\pi + 2\pi \sqrt{229}) \text{ cm}^2$
Factor out $2\pi$ :
$A_{total} = 2\pi (42 + \sqrt{229}) \text{ cm}^2$
Calculate the numerical value (using $\pi \approx 3.14159$ and $\sqrt{229} \approx 15.1327$ ).
$A_{total} \approx 2 \times 3.14159 \times (42 + 15.1327)$ $A_{total} \approx 6.28318 \times 57.1327$ $A_{total} \approx 359.08 \text{ cm}^2$
The total surface area of the rocket is $\boxed{2\pi (42 + \sqrt{229}) \text{ cm}^2 \approx 359.08 \text{ cm}^2}$ .

To find the total surface area of the rocket, we need to calculate the area of its exposed surfaces. The rocket is composed of a cylinder and a cone. The exposed surfaces are:

Step-by-step solution

Calculate the slant height ( $l$ ) of the cone.

Calculate the area of the circular base of the cylinder.

Calculate the lateral surface area of the cylinder.

Calculate the lateral surface area of the cone.

Calculate the total surface area ( $A_{total}$ ) by summing the areas of the exposed surfaces.

Calculate the numerical value (using $\pi \approx 3.14159$ and $\sqrt{229} \approx 15.1327$ ).

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More Mathematics Questions

To find the total surface area of the rocket, we need to calculate the area of its exposed surfaces. The rocket is composed of a cylinder and a cone. The exposed surfaces are:

Step-by-step solution

Calculate the slant height (lll) of the cone.

Calculate the area of the circular base of the cylinder.

Calculate the lateral surface area of the cylinder.

Calculate the lateral surface area of the cone.

Calculate the total surface area (AtotalA_{total}Atotal​) by summing the areas of the exposed surfaces.

Calculate the numerical value (using π≈3.14159\pi \approx 3.14159π≈3.14159 and 229≈15.1327\sqrt{229} \approx 15.1327229​≈15.1327).

Still stuck on this one?

More Mathematics Questions

Calculate the slant height ( $l$ ) of the cone.

Calculate the total surface area ( $A_{total}$ ) by summing the areas of the exposed surfaces.

Calculate the numerical value (using $\pi \approx 3.14159$ and $\sqrt{229} \approx 15.1327$ ).