A right cone has slant height 35
centimetres and surface area 450𝜋 square centimetres. What is the radius of its base?
We recall here that the surface
area of a cone is equal to 𝜋𝑟𝑙 plus 𝜋𝑟 squared. 𝜋𝑟𝑙 is equal to the curved or
lateral surface area of a cone. 𝜋𝑟 squared is equal to the base
area, as the base of a cone is a circle. Our value for 𝑟 is the radius of
the base, and 𝑙 is the slant height. We know that the total surface area
is 450𝜋. The slant height is 35
centimetres. Therefore, the curved surface area
is 35𝜋𝑟. The area of the base is 𝜋𝑟
squared. As 𝜋 is common to all three terms,
we can divide both sides of the equation by 𝜋. This gives us 450 is equal to 35𝑟
plus 𝑟 squared.
Subtracting 450 from both sides of
this equation will give us a quadratic equation equal to zero. 𝑟 squared plus 35𝑟 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 𝑟 plus 45 and 𝑟 minus 10.
In order to solve this equation
equal to zero, one of the parentheses must be equal to zero. Either 𝑟 plus 45 equals zero or 𝑟
minus 10 equals zero. Solving these two equations gives
us 𝑟 equals negative 45 or 𝑟 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 square
centimetres has a base radius of 10 centimetres.