A cone has a height of #18 cm# and its base has a radius of #7 cm#. If the cone is horizontally cut into two segments #6 cm# from the base, what would the surface area of the bottom segment be?

Answer 1

#(637/9 pi +(35sqrt373)/9 pi)# #cm^2#

Since height of cone is 18 and radius of base is 7, its slant height would be #sqrt (18^2 +7^2) = sqrt 373# cms

Area of the slant surface of the cone would be =#pi (7)sqrt373#= #7pi sqrt373# #cm^2.#

Now the cone is horizontally cut 6 cms from the base. it would form a smaller cone of vertical height 12 cms, as shown in the figure below. To find out the radius and slant height of the smaller cone, compare the side ratios of similar triangles AFC and AGE

#(AG)/(AF) = (GE)/(FC)= (AE)/(AC)#

#12/18=(GE)/7= (AE)/sqrt373#, which gives #GE=14/3, AE=(2sqrt373)/3#

Slant surface area of the smaller cone would, thus, be #pi (14/3)(2sqrt373)/3 = pi (28sqrt373)/9# #cm^2#

The bottom segment has been shown as a shaded portion in the figure. Its slant surface area would thus be #(7pisqrt373-pi(28sqrt373)/9)# #cm^2#

To get the total surface area of the bottom segment, area of its top and bottom plane surfaces need to added to the slant surface area.

Area of the top plane surface, which is a circle of radius #14/3#, would be #pi (14/3)^2= pi 196/9# #cm^2#

Area of bottom plane surface, which is a circle of radius 7 would be #49 pi#

Thus the total surface area of bottom segment would be #49 pi +196/9 pi +7 pisqrt373- pi(28sqrt373)/9#

= #(637/9 pi +(35sqrt373)/9 pi)# #cm^2#

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Answer 2

To find the surface area of the bottom segment of the cone, you first need to calculate the surface area of the whole cone, then subtract the surface area of the top segment (the smaller cone).

First, calculate the slant height (l) of the cone using the Pythagorean theorem: [l = \sqrt{r^2 + h^2}]

Where ( r = 7 , \text{cm} ) (radius) and ( h = 18 , \text{cm} ) (height).

Then, calculate the surface area of the whole cone using the formula: [A_{\text{whole}} = \pi r (r + l)]

Next, calculate the radius of the top segment. Since it's 6 cm from the base, the radius (r') of the top segment can be found using similar triangles: [ \frac{r'}{r} = \frac{h'}{h} ] Where ( h' = h - 6 = 18 - 6 = 12 ) cm.

Then, use the found ( r' ) to calculate the surface area of the top segment: [ A_{\text{top}} = \pi r' (r' + l') ] Where ( l' ) is the slant height of the top segment, calculated using the Pythagorean theorem.

Finally, subtract the surface area of the top segment from the surface area of the whole cone to get the surface area of the bottom segment.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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