A pyramid V-ABCD is cut from a cube of edge 12 in.,as shown in the figure. The

vertex V is the midpoint of an upper edge of the cube. Compute the lateral surface of the pyramid.

No figure. Cannot copy and paste here.

To compute the lateral surface area of the pyramid, we need to find the area of each triangular face and then sum them up.

Step 1: Find the base length (AB) and the slant height of the pyramid.
Since the figure shows that V is the midpoint of an upper edge of the cube, we can determine that line VD is perpendicular to the base ABCD. Therefore, triangle VAD is a right triangle.
The edge of the cube is given as 12 inches, so the base length of the pyramid is AB = 12 inches.
To find the slant height of the pyramid, we need to find the length of VA. Since V is the midpoint of the upper edge, VA can be calculated as half the diagonal of the cube's face.
The diagonal of a square face of a cube can be found using the Pythagorean theorem: diagonal = √(edge^2 + edge^2) = √(12^2 + 12^2) = √(144 + 144) = √288 = 12√2 inches.
So, VA = 1/2 * 12√2 = 6√2 inches.

Step 2: Calculate the area of each triangular face.
Since all the triangular faces of the pyramid are congruent, we just need to find the area of one triangle and multiply it by four to account for all four faces.
The area of a triangle can be calculated using the formula: Area = 1/2 * base * height.
In this case, the base (AB) is 12 inches, and the height (slant height VA) is 6√2 inches.
Therefore, the area of one triangular face is: 1/2 * 12 * 6√2 = 72√2 square inches.

Step 3: Compute the lateral surface area by multiplying the area of one triangular face by four.
Lateral Surface Area = 4 * 72√2 square inches = 288√2 square inches.