Question

find the volume of the following prisms and and pyramids:

Explanation:

Step1: Volume of the square pyramid

The formula for the volume of a pyramid is \(V=\frac{1}{3}Bh\), where \(B\) is the area of the base and \(h\) is the height. For a square base with side length \(s = 7\) yd, \(B=s^{2}=7\times7 = 49\) yd². The height \(h = 10\) yd. Then \(V=\frac{1}{3}\times49\times10=\frac{490}{3}=163\frac{1}{3}\) yd³.

Step2: Volume of the rectangular prism

The formula for the volume of a prism is \(V = Bh\), where \(B\) is the area of the base. For a rectangular base with length \(l = 3\) cm and width \(w = 2\) cm, \(B=l\times w=3\times2 = 6\) cm². The height of the prism \(h = 6\) cm. Then \(V=6\times6=36\) cm³.

Step3: Volume of the rectangular pyramid

For a rectangular base with length \(l = 22\) cm and width \(w = 12\) cm, \(B=l\times w=22\times12 = 264\) cm². The height \(h = 18\) cm. Using the pyramid volume formula \(V=\frac{1}{3}Bh\), we have \(V=\frac{1}{3}\times264\times18= 1584\) cm³.

Step4: Volume of the triangular pyramid

For a triangular base with base \(b = 6\) mi and height \(h_{base}=8\) mi, \(B=\frac{1}{2}\times b\times h_{base}=\frac{1}{2}\times6\times8 = 24\) mi². The height of the pyramid \(H = 10\) mi. Using \(V=\frac{1}{3}BH\), we get \(V=\frac{1}{3}\times24\times10 = 80\) mi³.

Step5: Volume of the triangular pyramid (another one)

For a triangular base with base \(b = 11\) km and height \(h_{base}=4\) km, \(B=\frac{1}{2}\times b\times h_{base}=\frac{1}{2}\times11\times4=22\) km². The height of the pyramid \(H = 10\) km. Then \(V=\frac{1}{3}\times22\times10=\frac{220}{3}\approx73.3\) km³.

Step6: Volume of the triangular prism

For a triangular base with base \(b = 3\) m and height \(h_{base}=4\) m, \(B=\frac{1}{2}\times b\times h_{base}=\frac{1}{2}\times3\times4 = 6\) m². The height of the prism \(H = 5\) m. Using the prism volume formula \(V = BH\), we have \(V=6\times5=30\) m³.

Answer:

Square pyramid: \(163\frac{1}{3}\) yd³, Rectangular prism: \(36\) cm³, Rectangular pyramid: \(1584\) cm³, Triangular pyramid (first): \(80\) mi³, Triangular pyramid (second): \(\frac{220}{3}\) km³, Triangular prism: \(30\) m³.