Question

all edges of a triangular prism measure √23 units. find the surface area s and volume v of the prism. the surface area of the prism is (simplify your answer. type an exact answer, using radicals as needed. use integers or fractions for any numbers in the expression.)

Explanation:

Step1: Find base - area of triangular base

The base is an equilateral triangle with side length \(a = \sqrt{23}\). The area formula for an equilateral triangle is \(A_{base}=\frac{\sqrt{3}}{4}a^{2}\). Substituting \(a = \sqrt{23}\), we get \(A_{base}=\frac{\sqrt{3}}{4}\times(\sqrt{23})^{2}=\frac{23\sqrt{3}}{4}\).

Step2: Calculate the volume of the prism

The volume formula for a prism is \(V = A_{base}\times h\). Since the height \(h=\sqrt{23}\), then \(V=\frac{23\sqrt{3}}{4}\times\sqrt{23}=\frac{23\sqrt{69}}{4}\).

Step3: Find the area of the three rectangular faces

Each rectangular face has dimensions \(\sqrt{23}\times\sqrt{23}\). The area of one rectangular face is \(A_{rect}=\sqrt{23}\times\sqrt{23}=23\). The total area of the three rectangular faces is \(3\times23 = 69\).

Step4: Calculate the total surface - area of the prism

The total surface - area \(S\) of the triangular prism is the sum of the areas of the two bases and the three rectangular faces. The area of two bases is \(2\times\frac{23\sqrt{3}}{4}=\frac{23\sqrt{3}}{2}\). So \(S=\frac{23\sqrt{3}}{2}+69\).

Answer:

Volume \(V = \frac{23\sqrt{69}}{4}\), Surface - area \(S=\frac{23\sqrt{3}}{2}+69\)