Confectionery 7318

The confectioner needs to carve a cone-shaped decoration from a ball-shaped confectionery mass with a radius of 25 cm. Find the radius of the base of the ornament a (and the height h). He uses as much material as possible is used to make the ornament.

Correct answer:

r =  23.57 cm
h =  33.33 cm
V =  19392.55 cm³

Step-by-step explanation:

$$\begin{gathered} G=25cm \\ V=\frac{1}{3}\pi r^{2}h \\ h=G+x=25+x \\ {G}^2=x^2+r^2 \\ {r}^2=G^2-x^2 \\ V=\frac{1}{3}\pi (G^2-x^2)(G+x) \\ V=\frac{1}{3}\pi (G^3-G^{2}x-G{x}^2-x^3) \\ {V}^{\prime }=\frac{1}{3}\pi (G^2-2Gx-3x^2) \\ {V}^{\prime }=0 \\ {G}^2-2Gx-3x^2=0 \\ -3x^2-50x+625=0 \\ 3x^2+50x-625=0 \\ a=3;b=50;c=-625 \\ D=b^2-4ac=50^2-4\cdot 3\cdot (-625)=10000 \\ D>0 \\ {x}_{1,2}=\frac{-b\pm \sqrt{D}}{2a}=\frac{-50\pm \sqrt{10000}}{6} \\ {x}_{1,2}=\frac{-50\pm 100}{6} \\ {x}_{1,2}=-8.333333\pm 16.666667 \\ {x}_1=8.333333333 \\ {x}_2=-25 \\ \text{ Factored form of the equation: } \\ 3(x-8.333333333)(x+25)=0 \\ h=G+x_1=25+8.3333333333333=33.333333333333cm \\ r=\sqrt{G^2-x_1^2}=23.57\text{cm} \end{gathered}$$

$h=25+8.3333333333333=33.33\text{cm}$

$V=\frac{1}{3}\pi r^{2}h=19392.55{\text{cm}}^3$

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