In the given figure, a tent is the
shape of a cylinder surmounted by a conical top of the same diameter. The height and diameter of the
cylindrical part are 2.1 meters and three meters, respectively, and the slant height
of the conical part is 2.8 meters. Determine the cost of the canvas
needed to make the tent if the canvas is available at a rate of 500 rupees per meter
squared. Use 𝜋 is equal to 22 over
Now for this question, we’ve been
asked to find the cost of the canvas needed to make the tent. In order to do so, we’ll first need
to find the area of the canvas that makes up the tent in meters squared. And we’ll then need to multiply
this by 500 to find the cost in rupees. The question tells us that the tent
has one part which is a cylinder and one part which is a cone.
Looking at the diagram, we can
understand that the canvas is split into two distinct parts. One is the curved surface area of
the conical top, and the other is the curved surface area of the cylinder. To answer this question, we can
first write out the formula for the curved surface area of a cone, which is given by
𝜋 times 𝑟 times 𝑙. Next, we can write out the formula
for the curved surface area of a cylinder, which is given by two 𝜋 times 𝑟 times
Let’s now define the terms written
in these equations. Firstly, we can see that both
equations contain the term 𝑟. This term 𝑟 is the radius of the
cone and the cylinder, respectively. Looking back at the question, we’re
told that both the cone and the cylinder have the same diameter, and this is three
meters. What this really means is the
circle that makes up the base of our cone and the end of our cylinder has a diameter
of three meters.
Now we also know that the radius of
any circle is half its diameter. This allows us to say that the
radius of our circle, and therefore the radius of both our cone and our cylinder, is
three over two meters. For now, we’re gonna choose to
leave this as a fraction as we may be able to cancel out some factors later.
Now that we have defined 𝑟, let us
look at the 𝑙 in the formula for the curved surface area of a cone. This 𝑙 represents the slant height
of the cone, and our question has given us this value as 2.8 meters. Finally, we look at the ℎ in the
formula for the curved surface area of a cylinder. This ℎ represents the height of the
cylinder, and our question has also given us this value, which is 2.1 meters.
Now that we’ve defined all our
terms, let’s work on finding the total area of the canvas. We can now remind ourselves that
the total area of the canvas is equal to the curved surface area of the cone added
to the curved surface area of the cylinder.
Before we substitute in our values,
let’s first see what happens when we add the two expressions together. Adding the two expressions together
gives us 𝜋𝑟𝑙 plus two 𝜋𝑟ℎ. We can see that both of our terms
have a factor of 𝜋𝑟. We’re therefore able to factor this
out, to obtain 𝜋𝑟 times 𝑙 plus two ℎ.
Let’s now substitute in the values
that we know. Firstly, we have that 𝑟, the
radius of the cone and the cylinder, is three over two. Next, we have that 𝑙, the slant
height of the cone, is 2.8. And lastly, we have that ℎ, the
height of the cylinder, is equal to 2.1. We can now work on our second set
of brackets, firstly by evaluating that two times 2.1 is equal to 4.2. We can then see that 2.8 plus 4.2
is a nice integer number, which is seven.
Now that we have simplified, we can
recall that the question gives us an approximate value for 𝜋, which is 22 divided
by seven. We can therefore also substitute
this value into our expression. After doing this, we see that we
can perform some more canceling, firstly by recognizing that we have a factor of one
over seven and seven, which evaluates to one. So we can cancel these two out. We also see that we have a factor
of 22 and a half, and 22 divided by two is 11. This leaves us with the expression
11 times three, which of course evaluates to 33. Since this represents the total
area of the canvas, we’ll recall that our units should be in meters squared. And we have therefore found that
the total area of the canvas is 33 meters squared.
After finding the total area of the
canvas, we must now calculate the cost of the canvas. The question tells us that the
canvas is available at a rate of 500 rupees per meter squared. This means that we must multiply
500 by each square meter of canvas that we have in the tent. Our calculation is therefore 500
Now we might find this calculation
easier if we say that 500 is equal to 1000 times a half. We can then evaluate that half of
33 is 16.5, and 1000 times 16.5 is 16500. In completing this step, we have
answered the question. And we should remember that our
units for cost in this case are rupees. We have therefore found that the
cost of the canvas to make this tent is 16500 rupees.