Question

a triangular prism has an equilateral base with side length x inches. the height of the prism is four times as long as the side length. what is the lateral area of the prism? 12x square inches 12x^2 square inches 36x square inches 36x^2 square inches

Explanation:

Step1: Determine the height of the prism

The side - length of the equilateral base is \(x\) inches. The height \(h\) of the prism is four times the side - length, so \(h = 4x\) inches.

Step2: Recall the formula for the lateral area of a triangular prism

The lateral area \(L\) of a triangular prism with perimeter of the base \(P\) and height \(h\) is \(L=Ph\). For an equilateral triangle base with side - length \(x\), the perimeter of the base \(P = 3x\) (since an equilateral triangle has 3 equal sides).

Step3: Calculate the lateral area

Substitute \(P = 3x\) and \(h = 4x\) into the formula \(L = Ph\). We get \(L=(3x)\times(4x)=12x^{2}\) square inches.

Answer:

12x² square inches