Katy 7

Katy ordered a cylinder-shaped cake with a volume of 15.7 litres. It consists of two layers. The volume of the upper layer is 4 times smaller than the volume of the lower layer. The height of both layers is the same and equals the radius of the upper layer. Katy cut the cake perpendicular to the base into 2 equal parts. What is the diameter of the lower layer of the cake? What is the height of the whole cake? What is the volume of one slice?

Final Answer:

d₂ =  40 cm
H =  20 cm
S =  599.8 cm²

Step-by-step explanation:

$$\begin{gathered} V=15.7\text{l}\to {\text{cm}}^3=15.7\cdot 1000{\text{cm}}^3=15700{\text{cm}}^3 \\ V=V_1+V_2 \\ {V}_1=V_2/4 \\ V=V_2/4+V_2 \\ 15700=V_2/4+V_2 \\ 5V_2=62800 \\ {V}_2=\frac{62800}{5}=12560 \\ {V}_2=12560 \\ {V}_1=V_2/4=12560/4=3140{\text{cm}}^3 \\ h=r_1 \\ {V}_1=\pi r_1^2\cdot h=\pi r_1^3 \\ {r}_1=\sqrt[3]{\frac{V_1}{\pi }}=\sqrt[3]{\frac{3140}{3.1416}}\doteq 9.9983\text{cm} \\ h=r_1=9.9983\doteq 9.9983\text{cm} \\ {V}_2=\pi r_2^2\cdot h \\ {r}_2=\sqrt{\frac{V_2}{\pi \cdot h}}=\sqrt{\frac{12560}{3.1416\cdot 9.9983}}\doteq 19.9966\text{cm} \\ {d}_2=2\cdot r_2=2\cdot 19.9966=40\text{cm} \end{gathered}$$

$H=2\cdot h=2\cdot 9.9983=20\text{cm}$

$S=h\cdot (2\cdot r_1)+h\cdot d_2=9.9983\cdot (2\cdot 9.9983)+9.9983\cdot 39.9932=599.8{\text{cm}}^2$

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