Question

assignment
date____ period____
find the volume of each figure. round your answers to the nearest hundredth, if necessary. leave your answers in terms of π for answers that contain π.
1)
2)
3)
4)
5)
6)

1. (v_{sphere}=\frac{4}{3}pi r^{3})

8)

Explanation:

1. For the first hexagonal - prism (problem 1):

• The area of a regular hexagon base \(A_{base}\) with side - length \(s = 7\) yd and apothem \(a = 6.1\) yd is given by \(A_{base}=3sa\).
• Calculate the base area:
• \(A_{base}=3\times7\times6.1 = 128.1\) square yards.
• The height of the prism \(h = 2\) yd.
• The volume of a prism \(V = A_{base}h\).
• \(V=128.1\times2=256.2\) cubic yards.

1. For the rectangular - prism (problem 2):

• The volume of a rectangular prism with length \(l = 7\) km, width \(w = 4\) km, and height \(h = 2\) km is given by \(V=lwh\).
• \(V = 7\times4\times2=56\) cubic kilometers.

1. For the rectangular - prism (problem 3):

• The volume of a rectangular prism with length \(l = 8\) yd, width \(w = 8\) yd, and height \(h = 4\) yd is given by \(V=lwh\).
• \(V=8\times8\times4=256\) cubic yards.

1. For the hexagonal - prism (problem 4):

• The area of a regular hexagon base \(A_{base}\) with side - length \(s = 4\) yd and apothem \(a = 2.8\) yd is given by \(A_{base}=3sa\).
• \(A_{base}=3\times4\times2.8 = 33.6\) square yards.
• The height of the prism \(h = 3\) yd.
• The volume of a prism \(V = A_{base}h\).
• \(V=33.6\times3 = 100.8\) cubic yards.

1. For the square - pyramid (problem 5):

• The area of the square base \(A_{base}=s^{2}\), where \(s = 5\) ft. So \(A_{base}=5^{2}=25\) square feet.
• The height of the pyramid \(h = 7\) ft.
• The volume of a pyramid \(V=\frac{1}{3}A_{base}h\).
• \(V=\frac{1}{3}\times25\times7=\frac{175}{3}\approx58.33\) cubic feet.

1. For the trapezoidal - prism (problem 6):

• The area of a trapezoid base \(A_{base}=\frac{(a + b)h}{2}\), where \(a = 5\) m, \(b = 8\) m, and \(h = 2.8\) m.
• \(A_{base}=\frac{(5 + 8)\times2.8}{2}=\frac{13\times2.8}{2}=18.2\) square meters.
• The length of the prism \(l = 3\) m.
• The volume of a prism \(V = A_{base}l\).
• \(V=18.2\times3 = 54.6\) cubic meters.

1. For the sphere (problem 7):

• The volume of a sphere with radius \(r = 4\) yd is given by \(V=\frac{4}{3}\pi r^{3}\).
• \(V=\frac{4}{3}\pi(4)^{3}=\frac{4}{3}\pi\times64=\frac{256\pi}{3}\approx268.08\) cubic yards.

1. For the cylinder (problem 8):

• The volume of a cylinder with radius \(r = 6\) m and height \(h = 6\) m is given by \(V=\pi r^{2}h\).
• \(V=\pi\times6^{2}\times6=\pi\times36\times6 = 216\pi\approx678.58\) cubic meters.

Answer:

1. \(256.2\) yd³
2. \(56\) km³
3. \(256\) yd³
4. \(100.8\) yd³
5. \(\frac{175}{3}\approx58.33\) ft³
6. \(54.6\) m³
7. \(\frac{256\pi}{3}\approx268.08\) yd³
8. \(216\pi\approx678.58\) m³