To find the surface area of a triangular prism with side lengths 3, 4, 4, and 5, we can use the formula:
Surface Area = 2(Area of the base) + (Perimeter of the base) x (Height of the prism)
First, let's find the area of the base. Since the base is a triangle, we can use the formula for the area of a triangle:
Area of base = (1/2) x base x height
The base of the triangle is the side with length 5, and the height can be found using the Pythagorean theorem:
height^2 = 4^2 - (3/2)^2
height^2 = 16 - 2.25
height^2 = 13.75
height = sqrt(13.75)
So the area of the base is:
Area of base = (1/2) x 5 x sqrt(13.75)
Area of base = 10.825
Next, let's find the perimeter of the base. Since the base is a triangle with side lengths 3, 4, and 5, the perimeter is:
Perimeter of base = 3 + 4 + 5
Perimeter of base = 12
Finally, let's find the height of the prism. We can use the Pythagorean theorem again:
height^2 = 4^2 - (3/2)^2
height^2 = 16 - 2.25
height^2 = 13.75
height = sqrt(13.75)
So the height of the prism is also sqrt(13.75).
Now we can plug these values into the formula for the surface area:
Surface Area = 2(Area of the base) + (Perimeter of the base) x (Height of the prism)
Surface Area = 2(10.825) + 12 x sqrt(13.75)
Surface Area = 21.65 + 41.4
Surface Area = 63.05
Therefore, the surface area of the triangular prism with side lengths 3, 4, 4, and 5 is 63.05 square units.