Answer to Question #318517 in Geometry for shibs

Question #318517

The base of a prism is a regular pentagon with one side measuring 15 cm. Its

Altitude is 32 cm. Find the following;

  1. Lateral surface Area in cm2.
  2. The total surface area in cm2.
  3. The volume of the prism in cm3.
1
Expert's answer
2022-03-29T12:18:06-0400

Let AB=BC=CD=DE=AEAB=BC=CD=DE=AE then AA1BB1=B=B1CC1=C=C1DD1=D=D1EE1 =E=E1AA1

Then the area of the upper and lower bases of the prism: Sbase=AB225+1054=22525+1054S\scriptsize{base} \normalsize =\frac{AB^2\sqrt{25+10\sqrt{5}}}4=\frac{225\sqrt{25+10\sqrt{5}}}4,

both bases: S=2S=22525+1052S^*=2S=\frac{225\sqrt{25+10\sqrt{5}}}2

1.

A side face has the area S=ABAAS=AB\cdot AA1 , all five side faces have the area S=5S=51532=2400S^{**}=5S=5\cdot15\cdot32=2400 cm2

2.

The total surface area: S=S+S=22525+1052+24003175,2148S=S^*+S^{**}=\frac{225\sqrt{25+10\sqrt{5}}}2+2400\thickapprox3175,2148 cm2

3.

The volume of the prism: V=SbaseAAV=S\scriptsize{base}\normalsize \cdot AA1 =22525+105432=180025+105=\frac{225\sqrt{25+10\sqrt{5}}}4\cdot32=1800\sqrt{25+10\sqrt{5}} 12387,4373\thickapprox12387,4373 cm3

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