A cone has a height of #12 cm# and its base has a radius of #6 cm#. If the cone is horizontally cut into two segments #9 cm# from the base, what would the surface area of the bottom segment be?

Answer 1

#357.253\ \text{cm}^2#

Radius #r# of new circular section of bottom segment cut horizontally, at a height #h=9\ cm# from base, from an original cone of height #H=12\ cm# & base radius #R=6\ cm# is given by using property of similar triangles as follows
#\frac{R-r}{h}=\frac{R}{H}#
#r=R(1-\frac{h}{H})#
#=6(1-9/12)#
#=1.5\ cm#

Now, surface area of bottom segment of original cone

#=\text{area of circular top of radius 1.5 cm}+\text{curved surface area of frustum of cone}+\text{area of circular base of radius 6 cm}#
#=\pir^2+\pi(r+R)\sqrt{h^2+(R-r)^2}+\piR^2#
#=\pi(1.5)^2+\pi(1.5+6)\sqrt{9^2+(6-1.5)^2}+\pi(6)^2#
#=357.253\ \text{cm}^2#
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Answer 2

To find the surface area of the bottom segment of the cone after it's horizontally cut 9 cm from the base, first, calculate the slant height of the smaller cone formed. Use the Pythagorean theorem: slant height = sqrt(height^2 + radius^2). Then, calculate the curved surface area of the smaller cone using the formula for the lateral surface area of a cone: π * radius * slant height. Finally, add the area of the circular base.

Surface Area = π * (radius) * (slant height) + π * (radius)^2

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Answer 3

To find the surface area of the bottom segment of the cone after it has been horizontally cut 9 cm from the base, we first need to determine the dimensions of the bottom segment.

The height of the bottom segment can be calculated as the total height of the cone (12 cm) minus the distance from the base where the cut was made (9 cm), which gives us a height of 3 cm for the bottom segment.

Next, we can calculate the radius of the bottom segment using similar triangles. Since the ratio of corresponding sides in similar triangles is constant, we have:

[ \frac{\text{radius of bottom segment}}{\text{radius of cone}} = \frac{\text{height of bottom segment}}{\text{height of cone}} ]

Substitute the values:

[ \frac{\text{radius of bottom segment}}{6 \text{ cm}} = \frac{3 \text{ cm}}{12 \text{ cm}} ]

Solving for the radius of the bottom segment:

[ \text{radius of bottom segment} = \frac{6 \text{ cm} \times 3 \text{ cm}}{12 \text{ cm}} = 1.5 \text{ cm} ]

Now that we have the radius and height of the bottom segment, we can calculate its surface area. The surface area of a cone is given by the formula:

[ \text{Surface Area} = \pi r \left( r + \sqrt{h^2 + r^2} \right) ]

Substitute the values for the radius (( r = 1.5 \text{ cm} )) and height (( h = 3 \text{ cm} )):

[ \text{Surface Area} = \pi \times 1.5 \text{ cm} \left( 1.5 \text{ cm} + \sqrt{(3 \text{ cm})^2 + (1.5 \text{ cm})^2} \right) ]

[ \text{Surface Area} = \pi \times 1.5 \text{ cm} \left( 1.5 \text{ cm} + \sqrt{9 \text{ cm}^2 + 2.25 \text{ cm}^2} \right) ]

[ \text{Surface Area} = \pi \times 1.5 \text{ cm} \left( 1.5 \text{ cm} + \sqrt{11.25 \text{ cm}^2} \right) ]

[ \text{Surface Area} = \pi \times 1.5 \text{ cm} \left( 1.5 \text{ cm} + 3 \text{ cm} \right) ]

[ \text{Surface Area} = \pi \times 1.5 \text{ cm} \times 4.5 \text{ cm} ]

[ \text{Surface Area} = 6.75 \pi \text{ cm}^2 ]

So, the surface area of the bottom segment of the cone is ( 6.75 \pi \text{ cm}^2 ).

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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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