Approximate the change in the lateral surface area(excluding the area of the base) of a right circular cone fixed height of h = 6m when its radius decreases from r=9m to r=8.9m (S=(pi)r sqrt(r^2+h^2)).
surface area= PI*r (sqrt(h^2+r^2)
A=PI (sqrt(r^4+(rh)^2)
dA=PI (1/2)(4r^3+2rh^2)/sqrt(r^4+(rh)^2)* dr
dr=-.1
r=9
h=6
To approximate the change in the lateral surface area of a right circular cone, we can calculate the difference between the lateral surface areas of the cone with the original radius and the cone with the new radius.
Given:
Initial radius, r₁ = 9m
Final radius, r₂ = 8.9m
Cone height, h = 6m
Lateral surface area formula: S = πr√(r^2 + h^2)
Step 1: Calculate the initial lateral surface area
Substitute the value of the initial radius (r₁ = 9) and the height (h = 6) into the lateral surface area formula:
S₁ = π * 9 * √(9^2 + 6^2)
S₁ ≈ 3.14 * 9 * √(81 + 36)
S₁ ≈ 3.14 * 9 * √(117)
S₁ ≈ 3.14 * 9 * 10.82
S₁ ≈ 325.44m²
Step 2: Calculate the final lateral surface area
Substitute the value of the final radius (r₂ = 8.9) and the height (h = 6) into the lateral surface area formula:
S₂ = π * 8.9 * √(8.9^2 + 6^2)
S₂ ≈ 3.14 * 8.9 * √(79.21 + 36)
S₂ ≈ 3.14 * 8.9 * √(115.21)
S₂ ≈ 3.14 * 8.9 * 10.73
S₂ ≈ 312.57m²
Step 3: Calculate the change in the lateral surface area
The change in the lateral surface area is the difference between S₂ and S₁:
Change in lateral surface area = S₂ - S₁
Change in lateral surface area ≈ 312.57 - 325.44
Change in lateral surface area ≈ -12.87m² (rounded to two decimal places)
Therefore, the approximate change in the lateral surface area (excluding the area of the base) of the right circular cone is approximately -12.87m².
To approximate the change in the lateral surface area of a cone, we need to calculate the initial lateral surface area and the final lateral surface area.
The formula for the lateral surface area of a cone is S = πr√(r^2 + h^2), where S is the lateral surface area, r is the radius, and h is the height.
First, let's calculate the initial lateral surface area:
S_initial = π(9)√((9^2) + (6^2))
= π(9)√(81 + 36)
= π(9)√(117)
≈ π(9)√(117) (approximately)
Next, let's calculate the final lateral surface area:
S_final = π(8.9)√((8.9^2) + (6^2))
= π(8.9)√(79.21 + 36)
= π(8.9)√(115.21)
≈ π(8.9)√(115.21) (approximately)
Now, we can find the approximate change in the lateral surface area by subtracting the initial lateral surface area from the final lateral surface area:
Change in S = S_final - S_initial
= π(8.9)√(115.21) - π(9)√(117)
≈ π(8.9)√(115.21) - π(9)√(117) (approximately)
So, to approximate the change in the lateral surface area of the cone, evaluate the expression π(8.9)√(115.21) - π(9)√(117).