From a solid cone, whose height is 16 cm and radius 12 cm, and conical cavity of height 3 cm and base radius 4 cm is hollowed out such that the bases of the cones form concentric circles. Find the total surface area of the remaining solid.
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Step-by-Step Solution
Step 1: Calculate slant heights
First, we need to find the slant heights of both the original cone and the conical cavity. The slant height (l) can be calculated using the Pythagorean theorem: l=r2+h2, where r is the radius and h is the height. For the larger cone, r1=12 cm and h1=16 cm. For the smaller conical cavity, r2=4 cm and h2=3 cm.
Step 2: Substitute values and calculate slant heights
Substitute the given values into the slant height formula. For the larger cone, l1=122+162=144+256=400=20 cm. For the smaller cone, l2=42+32=16+9=25=5 cm.
Step 3: Calculate areas
The total surface area of the remaining solid consists of three parts: the area of the circular base ring (the base of the large cone minus the base of the small cone), the curved surface area of the large cone, and the curved surface area of the hollowed-out small cone. The formula for the area of a circle is πr2 and for the curved surface area of a cone is πrl.
Step 4: Substitute values and calculate areas
Now, substitute the calculated radii and slant heights into the area formulas. The area of the base ring is π(122−42)=π(144−16)=128π cm2. The curved surface area of the large cone is π(12)(20)=240π cm2. The curved surface area of the small cone is π(4)(5)=20π cm2.
Step 5: Calculate total surface area
The total surface area of the remaining solid is the sum of these three calculated areas.
Step 6: Sum the areas
Adding the three areas together: 128π+240π+20π=388π cm2. If we approximate π≈3.14, the total surface area is approximately 388×3.14=1218.32 cm2.