Determine to the nearest hundredth the total surface area of the cone shown.
First, let’s recall the formula for calculating the surface area of a cone. It is 𝜋𝑟 squared plus 𝜋𝑟𝑙, where 𝑟 represents the radius of the cone and 𝑙 represents the slant height. The first term 𝜋𝑟 squared gives the area of the circle on the base of the cone, the second term 𝜋𝑟𝑙 gives the curved surface area of the cone, and together these give the total surface area.
We’ve been given the values of 𝑟 and 𝑙 in the diagram: 𝑟 is 19 centimetres and 𝑙 is 40 centimetres. Remember 𝑙 represents the slant height of the cone, not the perpendicular height, and it is the slant height that we’ve been given. So let’s substitute the values of 𝑟 and 𝑙 into our formula for the total surface area.
We have that the surface area is equal to 𝜋 multiplied by 19 squared plus 𝜋 multiplied by 19 multiplied by 40. Evaluating each of these constants gives 361 𝜋 plus 760 𝜋. Combining these two terms gives a total surface area of 1121 𝜋. The question remember has asked for the surface area to the nearest hundredth, so we need to evaluate this as a decimal and then round. The decimal is 3521.7253 and rounding this to the nearest hundredth will give us our answer: 3521.73, and the units of this surface area are centimetres squared.