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 12x² square inches 36x square inches 36x² square inches

Explanation:

Step1: Determine the height of the prism

The height $h$ of the prism is four - times the side length of the base. Given the side length of the base is $x$ inches, 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 an equilateral - base is given by the perimeter of the base times the height of the prism. The perimeter $P$ of an equilateral triangle with side length $x$ is $P=3x$ (since an equilateral triangle has three equal sides).

Step3: Calculate the lateral area

Using the formula $L = P\times h$, substitute $P = 3x$ and $h = 4x$ into it. Then $L=(3x)\times(4x)=12x^{2}$ square inches.

Answer:

12$x^{2}$ square inches