To find the volume of the pyramid, we need to know the formula for the volume of a pyramid, which is:
Volume of pyramid = 1/3 x area of base x height
We know that the base of the pyramid is the same as the triangular base given in the problem, with sides measuring 21, 17, and 10. We also know that the height of the pyramid is the same as the height of the prism.
To find the area of the base, we can use Heron's formula, which gives the area of a triangle when we know the lengths of its sides. Heron's formula is:
Area = sqrt[s(s - a)(s - b)(s - c)]
where a, b, and c are the sides of the triangle, and s is the semiperimeter (half the perimeter):
s = (a + b + c)/2
In this case, we have:
a = 21, b = 17, c = 10
s = (21 + 17 + 10)/2 = 24
Plugging these values into the formula, we get:
Area = sqrt[24(24 - 21)(24 - 17)(24 - 10)] = sqrt(24 x 3 x 7 x 14) = 84
Now we can plug in the area of the base and the height of the shape (which is given as the same for both the prism and the pyramid) into the formula for the volume of the pyramid:
Volume of pyramid = 1/3 x 84 x h
We know that the volume of the prism is 1,092 cubic units, and we can set this equal to the volume of the pyramid:
1,092 = 1/3 x 84 x h
Solving for h, we get:
h = 1,092 x 3 / (84 x 1) = 39
Now we can plug in h = 39, along with the area of the base (84), into the formula for the volume of the pyramid:
Volume of pyramid = 1/3 x 84 x 39 = 1,092 cubic units
Therefore, the volume of the pyramid is also 1,092 cubic units, option (1).