Response:
a. height of the pyramid: 5√7 cm
b. overall 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 measuring 10 cm and a side length of 15 cm. That side length serves as the hypotenuse of the right triangle formed when an altitude is drawn. The altitude length can be calculated using the Pythagorean theorem as...
h = √(15² -5²) = √200 = 10√2... cm
The lateral area equals the combined area of the four triangular faces, each one having this altitude and a base of 10 cm
LA = 4 × (1/2)bh = 2bh
LA = 2(10 cm)(10√2 cm) = 200√2 cm²
Additionally, the area of the square base is calculated as...
A = s² = (10 cm)² = 100 cm²
Thus, the total surface area of the pyramid can be expressed as...
A + LA = (100 +200√2) cm².... overall surface area
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a) The altitudes of opposite triangular faces along with a line drawn across the center of the base form another isosceles triangle. The height of this triangle corresponds to the pyramid's height. This height can also be derived with the Pythagorean theorem.
The altitude of the face acts as the hypotenuse of the right triangle, with half the base width being one side. The other side equates to the pyramid's height.
height = √((10√2)² -5²) = √175 = 5√7
The pyramid's height is 5√7 cm.
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c) The volume is determined using the formula...
V = (1/3)Bh
where B represents the base area (100 cm²) determined earlier, and h stands for the height (5√7 cm) calculated in section (a).
V = (1/3)(100 cm²)(5√7 cm) = (500/3)√7 cm³.... pyramid volume
