Sugar was traditionally produced and sold as sugarloaves, which are cones of sugar wrapped in paper. Find the total surface area of a cone-shaped sugarloaf with a base diameter of 8 centimeters and a slant height of 22 centimeters. Round answer to the nearest hundredth

A = 326,73 cm²

Step-by-step explanation:

The area of a circular cone is area of the base (A₁ ) plus area lateral ( area of a circular sector of radius the slant height )

Then we proceed to calculate the area of the base A₁

diameter of circular base is 8 cm then the radius is 4 cm and the area is:

A₁ = π*r²  =  3,14* (4)²

A₁ = 3,1416*16   =  50,2656 cm²

Now Lateral area of the cone (A₂) is equal to the area of a circular sector with radius the slant height. We will calculate it, taken into account that this circular sector is part of a circle of radius the slant height.

Between the area of circular sector with radius the slant height and the area of the circle with the same radius there is a linear relation. That is we can calculate area of a circular sector by rule of three as follows:

The length of the circular sector is the length of the circle of the base of the cone, that is:

L = 2*π*(4)

L = 2*3,1416*4

L = 25,1328 cm

Then we have a circular sector of length 25,1328 cm

The area of the circle of radius 22 cm is:

A(c)  = π*(22)²     ⇒   A(c)  = 1520,5344 cm²

And the length of this circle is:

L(c)  =  2*π*(22)     ⇒   138,2304 cm

Then we apply a rule of three

For a length of      138,2304 cm  ⇒⇒⇒  (area)  1520,5344 cm²

Then for a length of  25,1328 cm   ⇒⇒⇒(area)   A₂ (??)

Therefore:

A₂ =  (25,1328)*1520,5344)/ 138,2304

A₂ = 276,4608 cm²

Then total area of the cone is:

A = A₁  +  A₂

A =  50,2656  +276,4608

A  = 326,7264 cm²

Round answer  A = 326,73 cm²

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