To find the volume of a pyramid, we can use the formula: V = (1/3) * B * h, where B is the area of the base and h is the height.
In this case, the base of the pyramid is a triangle with sides of 4in, 5in, and 3in. To find the area of this triangle, we can use Heron's formula:
s = (4in + 5in + 3in)/2 = 6in
A = sqrt[s(s-4in)(s-5in)(s-3in)] = sqrt[6in(2in)(1in)(3in)] = sqrt[36in^2] = 6in*sqrt(2)
Now, we need to find the height of the pyramid. Let's draw a line from the top vertex of the pyramid down to the base, creating a right triangle.
The leg of this triangle opposite the 3in side of the base is the height we're looking for. Using the Pythagorean theorem, we can find this height:
h^2 = 5in^2 - (3in/2)^2 = 25in^2 - 9/4in^2 = 91/4in^2
h = sqrt(91/4)in
Now, we can plug in the values for B and h to find the volume:
V = (1/3) * 6in*sqrt(2) * sqrt(91/4)in
V = 9.46 cubic inches (rounded to two decimal places)
Therefore, the volume of the pyramid is approximately 9.46 cubic inches.