Allison designs fancy boxes to fill with chocolates. The boxes are in the shape of a right square pyramid as shown, where $$8 y$$ represents the length of one side of the base of the pyramid, and $$5 y$$ represents the height of one triangular face of the pyramid. ![](http://admin.getprephub.com/media/editor/gp18_20230225211328595723.png) The large size box must be designed to have a volume of 1,000 cubic centimeters. What dimensions for the length, in centimeters, of the base and the height of the triangular face, in centimeters, satisfy these constraints?

Given that, Base of the pyramid $$(b)=8 y$$, Slant height of the pyramid $$(I)=5 y$$, height of the pyramid=h (let) By Pythagoras theorem, $$(\frac{b}{2})^{2}+h^{2}=l^{2}$$ Or, $$(4 y)^{2}+h^{2}=(5 y)^{2}$$ Or, $$h^{2}=(5 y)^{2}-(4 y)^{2}=(3 y)^{2}$$ Or, $$h=3 y$$ Volume of the pyramid $$=\frac{1}{3} \times$$ area of the base $$\times$$ height $$ \text { Or, } \quad 1000=\frac{1}{3} \times(8 y)^{2} \times 3 y=64 y^{3} $$ $$\text { Or, } y^{3}=\frac{1000}{64}$$ Or, $${y}=\frac{10}{4}=\frac{5}{2}$$ Therefore base will be $$\frac{5}{2} \times 8=20 {~cm}$$ Height of the triangular face $$(5 y)=5 \times \frac{5}{2}=12.5 {~cm}$$

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Allison designs fancy boxes to fill with chocolates. The boxes are in the shape of a right square pyramid as shown, where $8 y$ represents the length of one side of the base of the pyramid, and $5 y$ represents the height of one triangular face of the pyramid.

The large size box must be designed to have a volume of 1,000 cubic centimeters. What dimensions for the length, in centimeters, of the base and the height of the triangular face, in centimeters, satisfy these constraints?

base $=20 {~cm}$ , height of the triangular face $=12.5 {~cm}$

base $=20 {~cm}$ , height of the triangular face $=7.5 {~cm}$

base $=12.5 {~cm}$ , height of the triangular face $=7.5 {~cm}$

None of the above

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