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 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
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.