A cone is made from a sector of a circle of radius [tex]HCm[/tex] and angle [tex]90^{\circ}[/tex]. Find the area of the curved surface of the cone.
Answer (2)
Calculate the arc length of the sector: A rc L e n g t h = 360 90 × 2 π × 14 = 7 π .
Determine the radius of the cone's base: r co n e = 2 π 7 π = 3.5 cm.
Identify the slant height of the cone: l = 14 cm.
Calculate the curved surface area of the cone: C u r v e d S u r f a ce A re a = π × 3.5 × 14 = 49 π square cm. Therefore, the curved surface area of the cone is 49 π .
Explanation
Problem Analysis We are given a cone formed from a sector of a circle with radius 14 cm and angle 9 0 ∘ . Our goal is to find the curved surface area of the cone.
Calculate Arc Length First, we need to find the arc length of the sector, which will become the circumference of the cone's base. The formula for arc length is: A rc L e n g t h = 360 an g l e × 2 π r where an g l e is the central angle of the sector and r is the radius of the original circle. In this case, an g l e = 9 0 ∘ and r = 14 cm. Therefore, A rc L e n g t h = 360 90 × 2 π × 14 = 4 1 × 2 π × 14 = 7 π So, the arc length is 7 π cm, which is approximately 21.99 cm.
Determine Cone's Base Radius Next, we determine the radius of the cone's base ( r co n e ) using the circumference. The circumference of the cone's base is equal to the arc length of the sector. The formula for the circumference of a circle is: C i rc u m f ere n ce = 2 π r co n e Since the circumference of the cone's base is 7 π cm, we have: 7 π = 2 π r co n e r co n e = 2 π 7 π = 2 7 = 3.5 So, the radius of the cone's base is 3.5 cm.
Identify Slant Height The radius of the original circle (14 cm) becomes the slant height ( l ) of the cone. So, l = 14 cm.
Calculate Curved Surface Area Now, we can calculate the curved surface area of the cone using the formula: C u r v e d S u r f a ce A re a = π r co n e l where r co n e is the radius of the cone's base and l is the slant height. In this case, r co n e = 3.5 cm and l = 14 cm. Therefore, C u r v e d S u r f a ce A re a = π × 3.5 × 14 = 49 π So, the curved surface area of the cone is 49 π square cm, which is approximately 153.94 square cm.
Final Answer Therefore, the curved surface area of the cone is 49 π square centimeters.
Examples
Cones are widely used in everyday life, from ice cream cones to traffic cones. Understanding how to calculate the surface area of a cone is useful in various applications. For instance, if you're designing a party hat in the shape of a cone, you'll need to know the surface area to determine how much material to use. Similarly, in construction, knowing the surface area of conical structures helps in estimating material costs and ensuring structural integrity. This knowledge is also applicable in fields like packaging design, where optimizing material usage is crucial for cost-effectiveness and sustainability.
The curved surface area of the cone made from a sector of a circle with radius 14 cm and angle 90° is calculated to be 49π square centimeters, which is roughly 153.94 square centimeters. This involves finding the arc length of the sector, determining the radius of the cone's base, and using the formula for the curved surface area. Following these steps clearly outlines the relationship between the parameters of the sector and the resulting cone.
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