Largest possible cone

It is necessary to make the largest possible cone from an iron rod in the shape of a prism with dimensions of 5.6 cm, 4.8 cm, and 7.2 cm. a) Calculate its volume. b) Calculate the waste.

Correct answer:

V =  43.4294 cm³
x =  150.1066 cm³

Step-by-step explanation:

$$\begin{gathered} a=5.6\text{cm} \\ b=4.8\text{cm} \\ c=7.2\text{cm} \\ {r}_1=\min (b/2,a/2)=\min (4.8/2,5.6/2)=\frac{12}{5}=2.4\text{cm} \\ {h}_1=c=7.2=\frac{36}{5}=7.2\text{cm} \\ {V}_1=\frac{1}{3}\cdot \pi \cdot r_1^2\cdot h_1=\frac{1}{3}\cdot \pi \cdot {\frac{12}{5}}^2\cdot h_1=\frac{1}{3}\cdot 3.1416\cdot 2.4^2\cdot 7.2\doteq 43.4294{\text{cm}}^3 \\ {r}_2=\min (a/2,c/2)=\min (5.6/2,7.2/2)=\frac{14}{5}=2.8\text{cm} \\ {h}_2=b=4.8=\frac{24}{5}=4.8\text{cm} \\ {V}_2=\frac{1}{3}\cdot \pi \cdot r_2^2\cdot h_2=\frac{1}{3}\cdot \pi \cdot {\frac{14}{5}}^2\cdot h_2=\frac{1}{3}\cdot 3.1416\cdot 2.8^2\cdot 4.8\doteq 39.4081{\text{cm}}^3 \\ {r}_3=\min (b/2,c/2)=\min (4.8/2,7.2/2)=\frac{12}{5}=2.4\text{cm} \\ {h}_3=a=5.6=\frac{28}{5}=5.6\text{cm} \\ {V}_3=\frac{1}{3}\cdot \pi \cdot r_3^2\cdot h_3=\frac{1}{3}\cdot \pi \cdot {\frac{12}{5}}^2\cdot h_3=\frac{1}{3}\cdot 3.1416\cdot 2.4^2\cdot 5.6\doteq 33.7784{\text{cm}}^3 \\ {V}_1>V_2>V_3 \\ V=V_1=43.4294=43.4294{\text{cm}}^3 \end{gathered}$$

$$\begin{gathered} K=a\cdot b\cdot c=5.6\cdot 4.8\cdot 7.2=193.536{\text{cm}}^3 \\ x=K-V=\frac{24192}{125}-V=193.536-43.4294=150.1066{\text{cm}}^3 \end{gathered}$$

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