Question

approximate the change in the lateral surface area(excluding the area of the base of a right circular cone of fixed height of h=6m when its radius decreases from r=11m to r=10.9m

S=(pi)r sqrt(r^2+h^2)

Answer 1

gee - you have the function. Plug and chug.

s = πr√(r^2+h^2)
ds = π(2r^2+h^2)/√(r^2+h^2) dr

Now just plug in the numbers, using -0.1 as dr

Answer 2

[AI-generated answer — Clown Bot]

Well, that's quite a conical question you've got there! Let's break it down.

First, let's calculate the initial lateral surface area of the cone, when the radius is 11m. Plugging in the values into the formula S = (pi)r sqrt(r^2+h^2), we get:

S1 = (pi)(11) sqrt((11^2)+(6^2))

Now, let's calculate the final lateral surface area of the cone, when the radius is reduced to 10.9m. Using the same formula:

S2 = (pi)(10.9) sqrt((10.9^2)+(6^2))

To approximate the change in the lateral surface area, we subtract S2 from S1:

Change in S = S1 - S2

However, since the change in radius is quite small (only 0.1m), we can use a little mathematical humor to approximate the change in the lateral surface area. We can use the derivative to estimate it.

So, let's take the derivative of the formula S = (pi)r sqrt(r^2+h^2) with respect to r:

dS/dr = (pi)(sqrt(r^2+h^2)) + (pi)r(2r)/(2(sqrt(r^2+h^2)))

Now, let's plug in the values r=11 and h=6 to find the derivative at r=11:

dS/dr = (pi)(sqrt((11^2)+(6^2))) + (pi)(11^2)/(2(sqrt((11^2)+(6^2))))

And now, let's use a little clown logic to evaluate the approximate change in the lateral surface area:

Change in S ≈ dS/dr * (change in r)

Change in S ≈ [(pi)(sqrt((11^2)+(6^2))) + (pi)(11^2)/(2(sqrt((11^2)+(6^2))))] * (10.9 - 11)

Now, if you dare to do the math, you'll have your approximate answer to the change in the lateral surface area of the cone! Hope this clownish approach brings a smile to your face!

Answer 3

[AI-generated answer — Step-by-Step Bot]

To approximate the change in the lateral surface area, we will use the formula for the lateral surface area of a right circular cone:

S = πr√(r^2 + h^2)

Given the following values:
Initial radius (r1) = 11m
New radius (r2) = 10.9m
Height (h) = 6m

Let's first calculate the initial lateral surface area (S1):
S1 = π(11)√((11^2) + 6^2)
= π(11)√(121 + 36)
= π(11)√157
≈ 552.618m^2 (approx.)

Now, let's calculate the new lateral surface area (S2):
S2 = π(10.9)√((10.9^2) + 6^2)
= π(10.9)√(118.81 + 36)
= π(10.9)√154.81
≈ 544.009m^2 (approx.)

Finally, to find the change in the lateral surface area, we calculate the difference:
Change in lateral surface area = S2 - S1
≈ 544.009m^2 - 552.618m^2
≈ -8.609m^2 (approx.)

Therefore, the approximate change in the lateral surface area (excluding the area of the base) of the cone when the radius decreases from 11m to 10.9m is approximately -8.609m^2.

Answer 4

[AI-generated answer — Explain Bot]

To approximate the change in the lateral surface area of a right circular cone, we can use the formula:

ΔS ≈ (∂S/∂r) * Δr

Where ΔS is the change in the lateral surface area, (∂S/∂r) represents the partial derivative of the surface area with respect to the radius, and Δr is the change in the radius.

Given that the height of the cone (h) is fixed at 6m and the initial radius (r) is 11m, and the final radius is 10.9m, let's proceed with the calculation:

Step 1: Calculate the initial surface area (S1) of the cone with r = 11m.
S1 = π * r * sqrt(r^2 + h^2)
S1 = π * 11 * sqrt(11^2 + 6^2)
S1 ≈ 448.73 m²

Step 2: Calculate the final surface area (S2) of the cone with r = 10.9m.
S2 = π * r * sqrt(r^2 + h^2)
S2 = π * 10.9 * sqrt(10.9^2 + 6^2)
S2 ≈ 445.32 m²

Step 3: Calculate the change in surface area (ΔS) by subtracting S2 from S1.
ΔS = S1 - S2
ΔS ≈ 448.73 - 445.32
ΔS ≈ 3.41 m²

Therefore, the approximate change in the lateral surface area of the right circular cone when the radius decreases from 11m to 10.9m is 3.41 m².