Question

surface area = ____ 16 ft 14 ft 5 ft 3 in

Explanation:

Step1: Identify the shapes of the faces

The prism has 2 congruent triangular faces and 3 rectangular faces.

Step2: Calculate area of triangular faces

The base of the triangle is 5 ft and the height (slant - height of the prism) is 16 ft. The area of a triangle is $A_{triangle}=\frac{1}{2}\times base\times height$. So, $A_{triangle}=\frac{1}{2}\times5\times16 = 40$ square feet. The total area of the two triangular faces is $2\times40=80$ square feet.

Step3: Calculate area of rectangular faces

There are three rectangular faces.

• One face has dimensions 5 ft by 14 ft, its area $A_1 = 5\times14=70$ square feet.
• Another face has dimensions 16 ft by 14 ft, its area $A_2=16\times14 = 224$ square feet.
• The third face has dimensions (using the Pythagorean - theorem to find the third side of the base triangle: $\sqrt{16^{2}-(\frac{5}{2})^{2}}=\sqrt{256 - \frac{25}{4}}=\sqrt{\frac{1024 - 25}{4}}=\sqrt{\frac{999}{4}}\approx15.8$ ft, but we can also use the fact that the three - rectangle method: the third rectangle has dimensions related to the perimeter of the base triangle. The third rectangle has dimensions such that its area $A_3$: The third rectangle has dimensions 14 ft by the hypotenuse of the base right - triangle. Using the Pythagorean theorem for the base triangle with legs 5 ft and $\sqrt{16^{2}-(\frac{5}{2})^{2}}$, but an easier way is to note that the three rectangles' areas: The third rectangle has dimensions such that its area $A_3$: The perimeter of the base triangle times the height of the prism. The perimeter of the base triangle is $5 + 16+\sqrt{16^{2}-(\frac{5}{2})^{2}}$. Another way: The three rectangles' areas: The first rectangle has sides 5 and 14, the second has sides 16 and 14, and the third: we know that the lateral surface area of a triangular prism is the perimeter of the base times the height of the prism. The perimeter of the base triangle with sides 5, 16 and $\sqrt{16^{2}-(\frac{5}{2})^{2}}$ is approximately $5 + 16+15.8 = 36.8$ ft. The height of the prism is 14 ft. The sum of the areas of the three rectangles: $A_{rectangles}=(5 + 16+\sqrt{16^{2}-(\frac{5}{2})^{2}})\times14$. A more straightforward way: The three rectangles have areas:
• $A_1 = 5\times14=70$ square feet.
• $A_2 = 16\times14=224$ square feet.
• Let the third side of the base triangle be $s=\sqrt{16^{2}-(\frac{5}{2})^{2}}\approx15.8$ ft, and the area of the third rectangle $A_3=14\times\sqrt{16^{2}-(\frac{5}{2})^{2}}\approx14\times15.8 = 221.2$ square feet. The sum of the areas of the three rectangles is $A_{rectangles}=70 + 224+221.2=515.2$ square feet.

The total surface area of the prism is the sum of the areas of the two triangular faces and the three rectangular faces.
The total surface area $A = 80+515.2=595.2$ square feet.

Answer:

$595.2$ square feet