A conical mountain has a radius of 1.5 kilometers and a perpendicular height of 0.5 kilometers. Determine the lateral area of the mountain to one decimal place.
Let’s begin by sketching this mountain, which we’re told is conical, so it is a cone shape. We’re told that it has a radius of 1.5 kilometers, so that’s the radius of its circular base, and a perpendicular height of 0.5 kilometers. So the mountain looks a little like this. We’re asked to determine the lateral area of the mountain, which is the area of the cone’s curved surface. We can recall that the formula for calculating the lateral area of a cone is 𝜋𝑟𝑙, where 𝑟 represents the radius of the cone and 𝑙 represents its slant height. That’s the distance from the apex of the cone to any point on the circumference of the base. On our figure, that’s this distance here, which we haven’t been given.
We can work this out though if we recall that the base radius, the perpendicular height, and the slant height of a cone form a right triangle. The Pythagorean theorem tells us that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the two shorter sides. In this triangle, the hypotenuse is the side marked 𝑙 kilometers and the two shorter sides are of length 1.5 and 0.5 kilometers. So we have the equation 𝑙 squared is equal to 1.5 squared plus 0.5 squared.
We can now solve this equation to determine the slant height of the cone. Evaluating each of the squares first, we have 𝑙 squared is equal to 2.25 plus 0.25, which simplifies to 2.5. To solve for 𝑙, we need to find the square root of each side of the equation, taking only the positive value as 𝑙 is a length. We have then that 𝑙 is equal to the square root of 2.5. Now, this is approximately equal to 1.58. But as we want to use this value of 𝑙 in the next stage of our working, we’ll keep it in this exact form.
We now return to our formula for the lateral area of a cone, and we substitute 1.5 for the base radius and the square root of 2.5 for the value of the slant height. Evaluating this on a calculator gives 7.450 continuing. The question specifies that we should give our answer to one decimal place. As the digit in the second decimal place is a five, we round up to 7.5. As the units for the lengths in this question were kilometers, the units for the area will be square kilometers.
So by first applying the Pythagorean theorem to calculate the slant height of this cone and then reusing the formula for the lateral area of a cone, we found that the lateral area of this mountain to one decimal place is 7.5 square kilometers.