From a Solid cylinder whose height is 2.4 cm and diameter 1.4 cm, a conical cavity of the same height and same diameter is hollowed out. Find the total Surface area of the remaining solid to the nearest cm² .

From a Solid cylinder whose height is 2.4 cm and diameter 1.4 cm, a conical cavity of the same height and same diameter is hollowed out. Find the total Surface area of the remaining solid to the nearest cm².

Given:

• Height of the cylinder (h) = 2.4 cm
• Diameter of the cylinder (d) = 1.4 cm

Radius of the cylinder (r) = d/2 = 0.7 cm

Height of the cone (h) = Height of the cylinder (h) = 2.4 cm

Radius of the cone (r) = Radius of the cylinder (r) = 0.7 cm

Slant height of the cone (l) = √(r^2 + h^2) l = √(0.7^2 + 2.4^2) l = √(0.49 + 5.76) l = √6.25 = 2.5 cm

Now, we can calculate the total surface area (TSA) of the remaining solid using the formula:

TSA = πr(2h+l+r)

=22/7 × 0.7 cm ×(2 ×2.4 cm +2.5 cm +0.7 cm)

=2.2 cm × 8 cm =17.6 = 17.6cm²

Therefore, the correct total surface area of the remaining solid, rounded to the nearest cm^2, is 18 cm^2.

Lateral Surface and Total Surface Area of Cone

Lateral Surface Area:

The lateral surface area of a cone refers to the curved portion of the cone, excluding the circular base. It can be calculated using the formula:

Lateral Surface Area (LSA) = π * r * l

where:

• π (pi) is a mathematical constant approximately equal to 3.14159.
• r is the radius of the cone’s base.
• l is the slant height of the cone, which is the length of the straight line from the apex (tip) to a point on the base edge.

Total Surface Area:

The total surface area of a cone includes both the lateral surface area and the area of the circular base. It can be calculated using the formula:

Total Surface Area (TSA) = LSA + Area of Base

Since the base is a circle, its area can be calculated using the formula:

Area of Base = π * r^2

Therefore, the complete formula for the total surface area of a cone is:

TSA = π * r * l + π * r^2

Key Points:

• Remember that the slant height (l) is not the same as the height (h) of the cone, which is the perpendicular distance from the apex to the base.
• If you only have the height (h) and radius (r), you can find the slant height (l) using the Pythagorean theorem: l^2 = h^2 + r^2.
• These formulas assume that the cone’s base is perfectly circular. If the base is elliptical, the calculations become more complex.

Example:

Suppose a cone has a radius of 5 cm and a slant height of 12 cm.

• Lateral Surface Area = π * 5 cm * 12 cm ≈ 188.5 cm²
• Area of Base = π * (5 cm)^2 ≈ 78.5 cm²
• Total Surface Area = 188.5 cm² + 78.5 cm² ≈ 267 cm²