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Regular pentagons and prisms

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1. Volume of a regular pentagonal prism ●○○○○ 10 ptsA right prism has regular pentagons as its bases. The side length of each pentagonal base is 4 cm and the height of the prism is 11 cm. Find the volume of the prism in cubic cm. Use the area of a regular pentagon $\approx 1.720477\,s^2$ for side length $s$. Round to two decimal places.
2. Total surface area of a pentagonal prism ●○○○○ 10 ptsA regular pentagonal prism has a base side length of 6 centimeters, a base apothem of 4 centimeters, and a height of 5 centimeters. Find its total surface area in square centimeters.
3. Side length from heptagon area ●○○○○ 10 ptsA regular heptagon has area $78.105$ square cm. Find its side length in cm, using the area formula $A \approx 3.633912444\,x^2$, where $x$ is the side length. Round to three decimal places.
4. Diagonal of a rectangular face ●●○○○ 20 ptsA regular right pentagonal prism has a length (the distance between its two pentagonal bases) of 18.9 cm and a volume of 452.7 cm$^3$. Find the diagonal length of one of its rectangular faces, in cm, rounded to the nearest millimetre (one decimal place). Use the area of a regular pentagon $\approx 1.720477\,s^2$ for side length $s$.
5. Area of a regular pentagon from perimeter ●●○○○ 20 ptsA regular pentagon has perimeter $85$ cm. Find its area in square centimetres. Round to the nearest square centimetre.
6. Apothem of a regular pentagon ●●○○○ 20 ptsA regular pentagon has sides of length 4 m. Find the length of its apothem, in metres. Round to two decimal places.
7. Minimal surface area for fixed volume ●●●○○ 50 ptsA regular right pentagonal prism has a volume of $278.3105$ cubic cm. Let $x$ be the side length of the regular pentagonal bases and $y$ the length (height) of the prism, in cm. Find the values of $x$ and $y$ that minimize the total surface area. Use the area of a regular pentagon $\approx 1.720477\,x^2$. Round each value to four decimal places.
8. Shortest network connecting regular pentagon vertices ●●●●○ 100 ptsA regular pentagon has side length 1 cm. Find the total length, in cm, of the shortest network of line segments that connects all five vertices of the pentagon. Give your answer in centimetres, rounded to two decimal places.

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