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A cone has a height of #5 cm# and its base has a radius of #5 cm#. If the cone is horizontally cut into two segments #1 cm# from the base, what would the surface area of the bottom segment be?

Answer 1

Ella Armstrong

#25pi+25sqrt2-(16pi+16sqrt2)=9pi+9sqrt2~=41 cm^2#

The surface area of a cone is:

#SA=pir(r+sqrt(h^2+r^2))#

We can calculate it for the cone at 5 cm height and at 4 cm height, subtract them, and thereby find the sliced off bit.

At h=5, r=5:

#SA=pi(5)(5+sqrt(5^2+5^2))=5pi(5+sqrt50)=25pi+5sqrt50=25pi+5sqrt25sqrt2=25pi+25sqrt2#

The angle of the cone doesn't change as we slice up the cone (it's 45 degrees), so the radius changes with the height. Therefore at h=4, r=4:

#SA=pi(4)(4+sqrt(4^2+4^2))=4pi(4+sqrt32)=16pi+4sqrt32=16pi+4sqrt16sqrt2=16pi+16sqrt2#

The difference is:

#25pi+25sqrt2-(16pi+16sqrt2)=9pi+9sqrt2~=41 cm^2#

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Answer 2

Claire Akin

The surface area of the bottom segment of the cone, after being horizontally cut 1 cm from the base, would be approximately 104.71 square centimeters.

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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.