Explicație pas cu pas:
a)
[tex]l = 6 \\ m = 6 \\[/tex]
[tex]a_{b} = \frac{l}{2} = 3 \\ [/tex]
[tex]h = \sqrt{ {m}^{2} - \left( \frac{l \sqrt{2} }{2} \right)^{2} } = \sqrt{ {6}^{2} - {(3 \sqrt{2})}^{2} } \\ = \sqrt{36 - 18} = \sqrt{18} = 3 \sqrt{2} \\ [/tex]
[tex]A_{b} = {l}^{2} = {6}^{2} = 36 \\ [/tex]
[tex]A_{l} = \frac{P_{b} \times a_{p}}{2} = \frac{4 \times l \times \sqrt{ {h}^{2} + a_{b}^{2} } }{2} \\ = 36 \sqrt{3} \\[/tex]
[tex]A_{t} = A_{b} + A_{l} = 36 + 36 \sqrt{3} = 36( \sqrt{3} + 1) \\ [/tex]
[tex]V = \frac{A_{b} \times h}{3} = \frac{36 \times 3 \sqrt{2} }{3} = 36 \sqrt{2} \\[/tex]
b)
[tex]l = 2 \times a_{b} = 2 \times 20 = 40 \\[/tex]
[tex]m = \sqrt{ {h}^{2} + {\left( \frac{l \sqrt{2} }{2} \right)}^{2} } = \sqrt{ {12}^{2} + {\left( \frac{40 \sqrt{2} }{2} \right)}^{2}} \\ = \sqrt{144 + 800} = \sqrt{944} = 4 \sqrt{59} [/tex]
[tex]a_{b} = 20[/tex]
[tex]h = 12[/tex]
[tex]A_{b} = {l}^{2} = {40}^{2} = 1600[/tex]
[tex]A_{l} = \frac{P_{b} \times a_{p}}{2} = \frac{4 \times l \times \sqrt{ {h}^{2} + a_{b}^{2} } }{2} = \frac{4 \times 40 \sqrt{ {12}^{2} + {20}^{2} } }{2} \\ = 80 \sqrt{544} = 320 \sqrt{34} [/tex]
[tex]A_{t} = A_{b} + A_{l} = 1600 + 320 \sqrt{34} \\ = 320(5 + \sqrt{34}) [/tex]
[tex]V = \frac{A_{b} \times h}{3} = \frac{1600 \times 12}{3} = 6400 \\[/tex]
c)
[tex]l = 10[/tex]
[tex]m = \sqrt{ {h}^{2} + {\left( \frac{l \sqrt{2} }{2} \right)}^{2} } = \sqrt{ {12}^{2} + {\left( \frac{10 \sqrt{2} }{2} \right)}^{2}} \\ = \sqrt{144 + 50} = \sqrt{194} [/tex]
[tex]a_{b} = \frac{l}{2} = \frac{10}{2} = 5 \\ [/tex]
[tex]h = \frac{3 \times V}{A_{b}} = \frac{3 \times 400}{100} = 12 \\ [/tex]
[tex]A_{b} = {l}^{2} = {10}^{2} = 100 [/tex]
[tex]A_{l} = \frac{P_{b} \times a_{p}}{2} = \frac{4 \times l \times \sqrt{ {h}^{2} + a_{b}^{2} } }{2} =\frac{4 \times 10 \times \sqrt{ {12}^{2} + 5^{2} } } {2} \\ = 20 \sqrt{144 + 25} = 20 \times 13 = 260 [/tex]
[tex]A_{t} = A_{b} + A_{l} = 100 + 260 = 360 \\ [/tex]
[tex]V = 400[/tex]
