Mack designed a water fountain with a square pyramid flowing into a cube. The edges of the bases of the pyramid and the cube have the same length and the heights of the pyramid and the cube are the same. Describe the relationship between the volume of the cube and the volume of the pyramid.

Volume of pyramid = 1/3 b2h
volume of  cube = b3

now, provided pyramid and cube have same edges and same heights.

thus, volume of pyramid = 1/3 b3
volume of cube = b3.

thus, ratio = 1/3b3/ b3 = 1/3.

Thus, volume of pyramid = 1/3volume of cube

$$V_{cube}=a^3\ \ \ and\ \ \ V_{pyramid}= \frac{1}{3} \cdot a^2\cdot h=\frac{1}{3}a^3\\ \\ \frac{V_{pyramid}}{V_{cube}} = \frac{\frac{1}{3}a^3}{a^3}= \frac{1}{3}\\ \\ \\A_{cube}=6a^2\\ \\h^2=a^2+( \frac{1}{2} a)^2\ \ \ \Rightarrow\ \ \ h^2= \frac{5a^2}{4}\ \ \ \Rightarrow\ \ \ h= \frac{a \sqrt{5} }{2}\\\\A_{pyramid}=a^2+4\cdot \frac{1}{2}ah=a^2+2a\cdot \frac{a \sqrt{5} }{2} =a^2(1+ \sqrt{5}) \\ \\ \frac{A_{pyramid}}{A_{cube}}= \frac{a^2(1+ \sqrt{5}) }{6a^2} = \frac{1+ \sqrt{5}}{6}$$

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