Answer to Question #149316 in Trigonometry for yui

Question #149316
PROBLEM 2. A block of wood is in the form of a right circular cone. The altitude is 12 cm and the radius of the base is 5 cm. A cylindrical hole of 5 cm is bored completely through the solid, the axis of the hole coinciding with the axis of the cone. Find the amount of wood left after the hole is bored.
FINAL ANSWER: _______________________________________
1
Expert's answer
2020-12-11T07:17:57-0500

"\\displaystyle\n\nh_1 = \\textsf{Height of cone}\\\\\n\nr_1 = \\textsf{Radius of cone}\\\\\n\n\nh_2 = \\textsf{Height of cylinder}\\\\\n\nr_1 = \\textsf{Radius of cylinder}\\\\\n\\textsf{Taking the ratios.}\\\\\n\\frac{h_1}{r_1} = \\frac{h_2}{r_2}\\\\\n\n\\frac{12}{5} = \\frac{5}{r_2}\\\\\n\nr_2 = \\frac{25}{12}\\\\\n\n\\begin{aligned}\n\\textsf{Volume of cone}\\, &= \\frac{\\pi {r_1}^2h_1}{3}\n\\\\&= \\frac{1}{3}\\times \\pi \\times 5^2 \\times 12 = 100\\pi\n\\end{aligned}\n\\\\\n\n\\begin{aligned}\n\\textsf{Volume of cylinder}\\,&=\\pi {r_2}^2 h_2\n\\\\&= \\pi \\times 5 \\times \\frac{25^2}{12^2} = \\frac{3125\\pi}{144}\\\\\n\\end{aligned}\\\\\n\n\\textsf{Amount of wood left}\\, = 100\\pi - \\frac{3125\\pi}{144} = \\frac{11275\\pi}{144} = 245.98\\, cm^3"


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