Chapter 12.2, Problem 44HP

To determine

To compare:The lateral areas of a rectangular prism and triangular prism.

Answer to Problem 44HP

The compression of two lateral areas is the ratio $L_R/L_T$ .

Explanation of Solution

Given data: $y$

The heights of a rectangular prism and triangular prism are same but base of triangular prism is isosceles triangle, the altitude of which is same as the height of the base of the rectangular prism.

Concept used:

The area of a triangle is $△\text{=}\frac{\text{1}}{\text{2}}\text{base×altitude}$ .

Calculations:

Suppose the dimensions of base of rectangular prism is $a\text{×}b$ and the each of equal sides of base of triangular prism is l as shown in figure here.

Now, the lateral areaof the rectangular prism $L_R=\text{2}\left(\text{area}△\text{ABZ+area}△\text{BZC}\right)$

  $\Rightarrow L_R=2\left(\frac{1}{2}\text{AB}\times \sqrt{h^{\text{2}}\text{+}{\left(\frac{a}{2}\right)}^{\text{2}}}\text{+}\frac{1}{2}\text{BC}\times \sqrt{h^{\text{2}}\text{+}{\left(\frac{b}{2}\right)}^{\text{2}}}\right)$

  $\Rightarrow L_R=\left(b\times \sqrt{h^{\text{2}}\text{+}{\left(\frac{a}{2}\right)}^{\text{2}}}\text{+}a\times \sqrt{h^{\text{2}}\text{+}{\left(\frac{b}{2}\right)}^{\text{2}}}\right)$

Now, the lateral area of the triangular prism $L_T=\left(\text{area}△\text{ZDO+area}△\text{ZDE+area}△\text{ZOE}\right)$

  $\Rightarrow L_T=\left(\frac{1}{2}\text{DE}\times \sqrt{h^{\text{2}}\text{+}{\left(\frac{a}{3}\right)}^{\text{2}}}\text{+}\frac{1}{2}\text{BC}\times \sqrt{h^{\text{2}}\text{+}{\left(p\right)}^{\text{2}}}+\frac{1}{2}\text{OD}\times \sqrt{h^{\text{2}}\text{+}{\left(p\right)}^{\text{2}}}\right)$

  $\Rightarrow L_T=\left(\frac{1}{2}\text{DE}\times \sqrt{h^{\text{2}}\text{+}{\left(\frac{a}{3}\right)}^{\text{2}}}\text{+}\frac{1}{2}l\times \sqrt{h^{\text{2}}\text{+}{\left(p\right)}^{\text{2}}}+\frac{1}{2}l\times \sqrt{h^{\text{2}}\text{+}{\left(p\right)}^{\text{2}}}\right)$

Where p is length of perpendicular from centroid of the triangle to equal sides.

The compression of two lateral areas is the ratio $L_R/L_T$ .

Conclusion:

The compression of two lateral areas is the ratio $L_R/L_T$ .