Question

a package is in the shape of a triangular prism. the bases are right triangles with perpendicular legs measuring 9 centimeters and 12 centimeters. the distance between the bases is 10 centimeters. what is the surface area of the triangular prism? 210 square centimeters 318 square centimeters 360 square centimeters 468 square centimeters

Explanation:

Step1: Calculate base - triangle area

The area of a right - triangle base $A_{base}=\frac{1}{2}\times9\times12 = 54$ square centimeters. Since there are 2 bases, the total area of the bases $A_{bases}=2\times54 = 108$ square centimeters.

Step2: Calculate the hypotenuse of the base triangle

Using the Pythagorean theorem $c=\sqrt{9^{2}+12^{2}}=\sqrt{81 + 144}=\sqrt{225}=15$ centimeters.

Step3: Calculate the areas of the rectangular faces

There are three rectangular faces.
The areas of the rectangular faces are:
Face 1 with dimensions 9 cm and 10 cm: $A_1=9\times10 = 90$ square centimeters.
Face 2 with dimensions 12 cm and 10 cm: $A_2=12\times10 = 120$ square centimeters.
Face 3 with dimensions 15 cm and 10 cm: $A_3=15\times10 = 150$ square centimeters.
The total area of the rectangular faces $A_{rectangles}=90 + 120+150=360$ square centimeters.

Step4: Calculate the total surface area

The total surface area of the triangular prism $A = A_{bases}+A_{rectangles}=108 + 360=468$ square centimeters.

Answer:

468 square centimeters