A cone has a height of #24 cm# and its base has a radius of #8 cm#. If the cone is horizontally cut into two segments #15 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 #8  cm#.  If the cone is horizontally cut into two segments #15 cm# from the base, what would the surface area of the bottom segment be?

Allison Allen · Accepted Answer

#:.color(purple)(=747.46cm^2# to the nearest # cm^2#

:.Pythagoras: #c^2=24^2+8^2#

#:.c=L=sqrt(24^2+8^2)#

#:. c=Lcolor(purple)(=25.298cm#

#:.24/8=tan theta=3.0=71^@33’54”#

:.#color(purple)(S.A.#= pir^2+pir*L#

:.S.A.#=3.141592654*8^2+pi*8*25.298#

:.S.A.#=201.062+635.808#

:.Total S.A.#color(purple)(=836.870cm^2#

#:.Cot 71^@33’54”*9=3.0cm=#radius of top part

:.Pythagoras: #c^2=9^2+3.0^2#

#:.c=L=sqrt(9^2+3.0^2)#

#:. c=Lcolor(purple)(=9.487cm# top part

:.S.A. top part#=pi*r^2+pi*r*L#

S.A. top part#:.3.141592654*3.0^2+pi*3.0*9.487#

S.A. top part#:.=28.274+89.413#

S.A. top part#:.color(purple)(=117.687cm^2#

:.S.A. Botom part#color(purple)(=836.870-117.687=719.183cm^2#

The total surface area of the bottom part got to include
the surface area of the circle of the top part.
:.S.A. Botom part:.=28.274+719.183=747.457 cm^2#

#:.color(purple)(=747.46cm^2# to the nearest 2 decimal places #

Parker Allen · Answer

undefined To find the surface area of the bottom segment of the cone after it's horizontally cut 15 cm from the base, we first need to find the slant height of the top part of the cone, which is $ \sqrt{24^2 + 8^2} = 8\sqrt{10} $ cm. Then, using this slant height, the radius of the top part of the cone becomes $ 8\sqrt{10} - 15 $ cm. Now, we calculate the surface area of the bottom segment using the formula for the surface area of a cone, subtracting the surface area of the top part from the whole cone's surface area. This gives us $ \pi(8^2) - \pi(8\sqrt{10} - 15)^2 $ cm$^2$. Simplifying this expression gives the surface area of the bottom segment of the cone.