Question

a prism has 2 congruent hexagonal bases like the one shown. each hexagon is made from 2 congruent isosceles trapezoids. the volume of the prism is 234 cubic units. what is the height of the prism? 3 units 6 units 8 units

Explanation:

1. First, find the area of the hexagonal - base:

• The hexagon is made up of 2 congruent isosceles trapezoids.
• The formula for the area of a trapezoid is \(A=\frac{(b_1 + b_2)h}{2}\), where \(b_1\) and \(b_2\) are the lengths of the parallel - sides and \(h\) is the height.
• For each trapezoid, \(b_1 = 5\), \(b_2=8\) (since \(4 + 4=8\)), and \(h = 3\).
• The area of one trapezoid \(A_1=\frac{(5 + 8)\times3}{2}=\frac{13\times3}{2}=\frac{39}{2}\) square units.
• The area of the hexagon (which is composed of 2 such trapezoids) \(A = 2\times\frac{39}{2}=39\) square units.

1. Then, use the volume formula for a prism:

• The volume formula for a prism is \(V=Ah\), where \(V\) is the volume, \(A\) is the area of the base, and \(h\) is the height of the prism.
• We know that \(V = 234\) cubic units and \(A = 39\) square units.
• Rearranging the formula \(h=\frac{V}{A}\).
• Substitute \(V = 234\) and \(A = 39\) into the formula: \(h=\frac{234}{39}=6\) units.

Answer:

6 units