Question Video: Finding the Base Radius of a Right Cone given Its Total Surface Area and Its Slant Height

A right cone has slant height 35 cm and surface area 450πœ‹ cmΒ². What is the radius of its base?

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Video Transcript

A right cone has slant height 35 centimetres and surface area 450πœ‹ square centimetres. What is the radius of its base?

We recall here that the surface area of a cone is equal to πœ‹π‘Ÿπ‘™ plus πœ‹π‘Ÿ squared. πœ‹π‘Ÿπ‘™ is equal to the curved or lateral surface area of a cone. πœ‹π‘Ÿ squared is equal to the base area, as the base of a cone is a circle. Our value for π‘Ÿ is the radius of the base, and 𝑙 is the slant height. We know that the total surface area is 450πœ‹. The slant height is 35 centimetres. Therefore, the curved surface area is 35πœ‹π‘Ÿ. The area of the base is πœ‹π‘Ÿ squared. As πœ‹ is common to all three terms, we can divide both sides of the equation by πœ‹. This gives us 450 is equal to 35π‘Ÿ plus π‘Ÿ squared.

Subtracting 450 from both sides of this equation will give us a quadratic equation equal to zero. π‘Ÿ squared plus 35π‘Ÿ minus 450 is equal to zero. We can solve this by factoring or factorising. We need to find two numbers that have a product of negative 450 and a sum of 35. 45 multiplied by negative 10 is negative 450. And 45 plus negative 10 is equal to 35. This means that our two brackets, or parentheses, will be π‘Ÿ plus 45 and π‘Ÿ minus 10.

In order to solve this equation equal to zero, one of the parentheses must be equal to zero. Either π‘Ÿ plus 45 equals zero or π‘Ÿ minus 10 equals zero. Solving these two equations gives us π‘Ÿ equals negative 45 or π‘Ÿ equals 10. The radius is a length and, therefore, cannot be negative. We can, therefore, conclude that a right cone with slant height 35 centimetres and surface area 450πœ‹ square centimetres has a base radius of 10 centimetres.

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