A cone has a height of #24 cm# and its base has a radius of #15 cm#. If the cone is horizontally cut into two segments #12 cm# from the base, what would the surface area of the bottom segment be?

Question

A cone has a height of #24  cm# and its base has a radius of #15  cm#.  If the cone is horizontally cut into two segments #12 cm# from the base, what would the surface area of the bottom segment be?

Allison Allen · Accepted Answer

Total Surface Area of the bottom segment of the cone

#A_(R-r) = color(purple)(1882.6)cm^3#

#r / R = h / H#

#r = (15 * (24-12))/24 = 7.5 cm#

Lateral surface are of a cone #L_R= pi R L#

Now we have to find L

#L = sqrt(R^2 + H^2) = sqrt(15^2 + 24^2) ~~ 28.3 cm#

#L_R = pi * 15 * 28.3 ~~ 1333.6 cm^2#

Similarly,#l = sqrt(12^2 + 7.5^2) ~~ 14.2 cm#

#l_r = pi r l = pi * 7.5 * 14.2 ~~ 334.6#

Area of base of a cylinder #A_R = pi R^2 = pi * 15^2 ~~ 706.9#

Similarly, #A_r = pi r^2 = pi * 7.5^2 ~~ 176.7#

Total surface area of the truncated cone

#A_(R-r) = L_R - l_r + A_R + A_r = 1333.6 - 334.6 + 706.9 + 176.7 = color(purple)(1882.6)cm^3#

Sadie Allen · Answer

22.1 SQ CM

Requires reworking based on the details given below

#A_T= A_B + A_b + L_R - L_r#

Where #A_T# is Total Surface Area of the bottom truncated cone, #A_B# is the area of the bottom circular base, #A_b# is the area of the circular base of top cut cone, #L_R# is the Lateral Surface Area of the whole cone and #l_r# is the Lateral Surface Area of the top half cone.

Evelyn Arboleda · Answer

undefined To find the surface area of the bottom segment of the cone, we need to calculate the area of the circular base and the lateral surface area of the frustum formed by the cut.

First, we find the slant height of the frustum using the Pythagorean theorem. Then, we calculate the lateral surface area of the frustum. Finally, we add the area of the circular base to get the total surface area of the bottom segment.

Let's denote:
- $ r_1 $ as the radius of the bottom segment
- $ r_2 $ as the radius of the top segment
- $ h_1 $ as the height of the bottom segment
- $ h_2 $ as the height of the top segment
- $ h $ as the total height of the cone

Given:
- $ r_1 = 15 $ cm (radius of the base)
- $ h_1 = 12 $ cm (height of the bottom segment)
- $ h = 24 $ cm (total height of the cone)

We can find $ r_2 $ using similar triangles:
$$ \frac{r_2}{r_1} = \frac{h_2}{h_1} $$
$$ \frac{r_2}{15} = \frac{24 - 12}{24} $$
$$ r_2 = 7.5 $$ cm

Now, we find the slant height ($ l_1 $) of the bottom segment using the Pythagorean theorem:
$$ l_1 = \sqrt{r_1^2 + h_1^2} $$
$$ l_1 = \sqrt{15^2 + 12^2} $$
$$ l_1 ≈ 19.21 $$ cm

Next, we calculate the lateral surface area ($ A_{\text{lateral}} $) of the frustum:
$$ A_{\text{lateral}} = \pi(r_1 + r_2)l_1 $$
$$ A_{\text{lateral}} = \pi(15 + 7.5)(19.21) $$
$$ A_{\text{lateral}} ≈ 1722.83 $$ sq. cm

Finally, we find the area of the circular base:
$$ A_{\text{base}} = \pi r_1^2 $$
$$ A_{\text{base}} = \pi (15)^2 $$
$$ A_{\text{base}} = 225\pi $$ sq. cm

The surface area of the bottom segment is the sum of the lateral surface area and the area of the circular base:
$$ A_{\text{bottom segment}} = A_{\text{lateral}} + A_{\text{base}} $$
$$ A_{\text{bottom segment}} ≈ 1722.83 + 225\pi $$ sq. cm