# Advanced Math If the height of a pyramid is increased by 60% and the length of an arm of a square is reduced by 35%, how much will the size of the pyramid increase or decrease?

Question
Upper Level Math
If the height of a pyramid is increased by 60% and the length of an arm of a square is reduced by 35%, how much will the size of the pyramid increase or decrease?

2020-12-28
The volume of both pyramids.
The volume of the square pyramid is $$\displaystyle{V}=\frac{{1}}{{3}}{a}^{{2}}{h}$$ where a is a one side length of the square of base of the pyramid.
suppose assume that a=10, h=10.
The volume of the pyramid is
$$\displaystyle{V}=\frac{{1}}{{3}}{10}^{{2}}{10}$$
$$\displaystyle=\frac{{1}}{{3}}{\left({1000}\right)}$$
=333.333
suppose height increase 60% and reduce the length of an arm of a square is reduced by 35%.
$$\displaystyle{h}_{{1}}={10}+\frac{{60}}{{100}}{\left({1}\right)}$$
=16
$$\displaystyle{a}_{{1}}={10}-\frac{{35}}{{100}}{\left({10}\right)}$$
=6.5
Volume of new pyramid is computed as follows,
$$\displaystyle{V}_{{1}}=\frac{{1}}{{3}}{\left({6.5}\right)}^{{216}}=\frac{{1}}{{3}}{\left({42.25}\right)}{16}$$
=225.333
Here, the new volume decreases, the difference of volume is,
$$\displaystyle{V}-{V}_{{1}}={333.333}-{225.333}$$
=108
Thus, the new volume decreases 108 units.
Compute the size of pyramid decreases
$$\displaystyle\frac{{{V}-{V}_{{1}}}}{{V}}\times{100}=\frac{{108}}{{333.333}}\times{100}$$
$$\displaystyle={0.3240003}\times{100}$$
=32.4
Therefore, the size of new pyramid decreases 32.4%.

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