A cone has a height of #7 cm# and its base has a radius of #2 cm#. If the cone is horizontally cut into two segments #3 cm# from the base, what would the surface area of the bottom segment be?

Question

A cone has a height of #7  cm# and its base has a radius of #2  cm#.  If the cone is horizontally cut into two segments #3 cm# from the base, what would the surface area of the bottom segment be?

Allison Allen · Accepted Answer

Total Surface Area of the truncated cone is #color(green)(A_T = 47.4757#

#R / H = r / (H - h)#

#r = (R * (H-h)) / H = (2 * (7-3)) / 7 = 8/7#

Lateral surface area of full cone

#A_f = pi R * L_1 = pi R sqrt(R^2 + H^2)#

#A_f = pi 2 * sqrt(2^2 + 7^2) = 45.7423#

Lateral Surface Area of top porion of the cut cone

#A_c = pi r L_2 = pi * (8/7) * sqrt((8/7)^2 + (7-3)^2) = 14.9363#

Lateral Surface area of the truncated cone

#A_t = A_f - A_c = 45.7423 - 14.9363 = color(brown)(30.806#

Base area of full cone

#A_B = pi R^2 = pi * 2^2 = color(brown)(12.5664#

Base area of top portion of the cut cone (top surface area of the frustum of the cone)

#A_b = pi * r^2 = pi * (8/7)^2 = color(brown)(4.1033)#

Total Surface Area of the truncated cone is

#A_T = A_t + A_B + A_b = 30.806 + 12.5664 + 4.1033) = color(green)(47.4757)#

Sadie Allen · Answer

undefined To find the surface area of the bottom segment of the cone after it's horizontally cut 3 cm from the base, you can follow these steps:

1. Find the slant height of the bottom segment using the Pythagorean theorem: 
   $l = \sqrt{r^2 + h^2}$
   $l = \sqrt{2^2 + 3^2}$
   $l = \sqrt{4 + 9}$
   $l = \sqrt{13}$ cm

2. Find the radius of the bottom segment using similar triangles:
   $\frac{r_{\text{segment}}}{r_{\text{cone}}} = \frac{h_{\text{segment}}}{h_{\text{cone}}}$
   $\frac{r_{\text{segment}}}{2} = \frac{4}{7}$
   $r_{\text{segment}} = \frac{8}{7}$ cm

3. Calculate the surface area of the bottom segment using the formula for the lateral surface area of a cone:
   $A_{\text{segment}} = \pi r_{\text{segment}} l$
   $A_{\text{segment}} = \pi \times \frac{8}{7} \times \sqrt{13}$ square cm

4. Calculate the numerical value of the surface area of the bottom segment using a calculator.

Therefore, the surface area of the bottom segment of the cone would be approximately $28.18$ square cm (rounded to two decimal places).