Answer:
2. a. The volume of the pyramid is approximately 363.62 cm³
b. The depth of the solution in the dish is approximately 0.15046 cm
Explanation:
2. a. A right square pyramid is a pyramid with the vertex at the top directly above the center of the base
The area of the base of the right square pyramid = 100 cm²
The area of one of the face = 60 cm²
∴ The length of the side of the base, s = √(100 cm²) = 10 cm
The area of the face = 1/2 × s × l
Where;
l = The slant height
∴ 1/2 × 10 cm × l = 60 cm²
l = 60 cm²/(5 cm) = 12 cm
l = 12 cm
The height of the pyramid, h = √(l² - (s/2)²)
∴ h = √(12² - 5²) = √(119)
The volume of the pyramid = 1/3 × Base Area × Height
∴ The volume of the pyramid, V = 1/3 × 100 cm² × √(119) cm ≈ 363.62 cm³
The volume of the pyramid, V ≈ 363.62 cm³
b. The initial level of the solution in the pyramid = The pyramid was filled with the solution
∴ The initial level of the solution in the pyramid = The height of the pyramid
The radius of the cylindrical dish into which the solution is poured, r = 4 cm
The level by which the liquid drops in the pyramid after pouring = 3 cm
Therefore, by proportion of volume, we have;
Let V₁ and 'V' represent the volume that was drained out the vertex into the cylindrical dish and the volume of the whole pyramid respectively
Therefore;
V₁/V = (3/(√119))³
V₁ = V × (3/(√119))³
V = 1/3 × 100 cm² × √(119) cm
∴ V₁ = 1/3 × 100 cm² × √(119) cm × (3/(√119))³ = 100 cm² × 1 cm × 3²/(119) = 900/119 cm³
The volume that was drained out the vertex, V₁ = 900/119 cm³
The volume of the cylindrical dish,
= π·r²·h
Where h = The depth of the solution in the dish
∴ When
= V₁, and r = 4 cm, we have;
π × 4² × h = 900/119
h = (900/119)/(π × 4²) ≈ 0.15046
The depth of the solution in the dish, h ≈ 0.15046 cm.