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 - area

The area of a right - triangle base with legs $a = 9$ cm and $b = 12$ cm is $A_{base}=\frac{1}{2}\times a\times b$.
$A_{base}=\frac{1}{2}\times9\times12 = 54$ square centimeters. Since there are 2 bases, the total area of the bases is $2\times A_{base}=2\times54 = 108$ square centimeters.

Step2: Calculate hypotenuse of the base triangle

Using the Pythagorean theorem $c=\sqrt{a^{2}+b^{2}}$, where $a = 9$ cm and $b = 12$ cm.
$c=\sqrt{9^{2}+12^{2}}=\sqrt{81 + 144}=\sqrt{225}=15$ cm.

Step3: Calculate lateral - face areas

The three lateral - face areas are:
One with dimensions $9\times10$, area $A_1=9\times10 = 90$ square centimeters.
One with dimensions $12\times10$, area $A_2=12\times10 = 120$ square centimeters.
One with dimensions $15\times10$, area $A_3=15\times10 = 150$ square centimeters.
The total lateral - face area is $A_{lateral}=A_1 + A_2+A_3=90 + 120+150=360$ square centimeters.

Step4: Calculate total surface area

The total surface area $A$ of the triangular prism is the sum of the area of the bases and the lateral - face area.
$A=A_{lateral}+2\times A_{base}=360 + 108=468$ square centimeters.

Answer:

D. 468 square centimeters