Answer:
a. pyramid height: 5√7 cm
b. total area: (100 +200√2) cm²
c. volume: (500/3)√7 cm³
Step-by-step explanation:
b) Each triangular face is an isosceles triangle with a base of 10 cm and a side length of 15 cm. That side length is the hypotenuse of the right triangle formed when an altitude is drawn. The length of the altitude is given by the Pythagorean theorem as ...
h = √(15² -5²) = √200 = 10√2 . . . cm
The lateral area is the area of the four triangular faces, each with this altitude and a base length of 10 cm
LA = 4 × (1/2)bh = 2bh
LA = 2(10 cm)(10√2 cm) = 200√2 cm²
Of course, the area of the square base is ...
A = s² = (10 cm)² = 100 cm²
So, the total surface area of the pyramid is ...
A + LA = (100 +200√2) cm² . . . . total surface area
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a) The altitudes of opposite triangular faces, together with a line across the center of the base, form another isosceles triangle. The height of this triangle is the height of the pyramid. Again, its height can be found using the Pythagorean theorem.
The altitude of the face is the hypotenuse of the right triangle, and half the width of the base is one side of the triangle. The other side is the height of the pyramid.
height = √((10√2)² -5²) = √175 = 5√7
The height of the pyramid is 5√7 cm.
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c) The volume is given by the formula ...
V = (1/3)Bh
where B is the area of the base (100 cm²) that we found above, and h is the height (5√7 cm) found in part (a).
V = (1/3)(100 cm²)(5√7 cm) = (500/3)√7 cm³ . . . . pyramid volume
