Video: Finding the Total Surface Area of Two Right Circular Cones with a Common Base to Solve a Real-Life Application

The buoy below is made of two right circular cones on a common base of radius 27 cm. If the cost of an erosion-resistant coat is 300 LE per square meter, find to the nearest tenth, the cost of painting the buoy.

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Video Transcript

The buoy below is made of two right circular cones on a common base of radius 27 centimeters. If the cost of an erosion-resistant coat is 300 Egyptian pounds per square meter, find to the nearest tenth the cost of painting the buoy.

So here we have a buoy made out of two cones. To find the cost of painting the buoy, that means we’ll need to work out the surface area of it. In this situation, when we’re finding the surface area of these two cones, we don’t need to worry about this circular section at the center because it won’t be painted. We’re therefore interested in the lateral surface area as we’re simply interested in the surface area of the shape excluding the base.

The formula for the lateral surface area of a cone is 𝜋 times the radius times 𝐿, which is the slant height of the cone. We can begin by finding the lateral surface area of the top cone. Here the slant height is 62 centimeters, and we’re not shown the radius on the diagram. But we were told that these two cones have a common base of radius 27 centimeters.

So we calculate 𝜋 times 27 times 62. Using a calculator, we can evaluate this as 5259.026102 and continuing. And because it’s a surface area, our units will be in squared units, which is square centimeters here.

Next, we calculate the lateral surface area of the lower cone. But we may notice that we have a problem. The length given as 71 centimeters is not the slant height of the cone, but instead the perpendicular height. In order to calculate the slant height, we’ll need to use the Pythagorean theorem. We can take a closer look at the triangle formed within the cone. And we know that there will be a right angle as this cone is a right circular cone.

We know that the height of the cone was 71 centimeters. And the radius, the top length, will be 27 centimeters. We can represent the slant height or the hypotenuse of this triangle by 𝐿. The Pythagorean theorem tells us that the square of the hypotenuse is equal to the sum of the squares on the other two sides. Substituting in our values, we’ll have 𝐿 squared equals 27 squared plus 71 squared. So we have 729 plus 5041, which is equivalent to 5770.

To find 𝐿, we take the square root of both sides. So we’ll have 𝐿 is equal to the square root of 5770. We can keep this value in the square-root form as we’ll use it in the lateral surface area calculation. Inputting this value into our lower cone lateral surface area, we’ll calculate 𝜋 times 27 times the square root of 5770, which evaluates as 6443.198979 and so on square centimeters.

To find the total surface area then, we add together the lateral surface areas of our two cones, which gives us a value of 11702.22508 and so on square centimeters. Here we can see that our surface area is in square centimeters. But the cost of the erosion coat is given in a cost per square meters. To change a value given in square centimeters into one given in square meters, we divide by 10000. So our surface area will be 1.1702 square meters.

Now that we know our surface area in square meters, we’re told that the cost is 300 Egyptian pounds per square meter. We can calculate the total cost then by multiplying 300 by 1.1702, which gives a value of 351.066 Egyptian pounds. And as our final answer is to be to the nearest tenth, we’ll have 351.1 Egyptian pounds.

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