Question 822055: You are to design a fountain for the garden of one of Emily’s customers.
• The fountain will have a cylindrical base with a cone on top.
• The cylindrical base will have a diameter of 1 m.
• The fountain is to be made of concrete.
• The entire fountain is to be coated with protective paint.
a) Make a sketch of your design, showing all
dimensions.
b) How much concrete is needed to make the
fountain?
c) What is the surface area that needs to be
painted?
d) Concrete costs $100/m3. Each litre of
protective paint costs $17.50 and covers
5 m2. Find the total cost of the materials
needed to make the fountain.
Answer by KMST(5328)   (Show Source):
You can put this solution on YOUR website! Not enough data, and not reasonable data.
You must be missing some details of the problem.
A 1 meter wide cylindrical base means a monstrously large fountain. It would be appropriate for a city square, but not for the garden of any citizen.
A 1 meter diameter would be appropriate for the top cone of the fountain in someone's garden,
but even assuming 1 meter diameter and 1 meter base height, there are other design issues that are not specified.
Is the top cone part a solid cone?
If there is a cavity in the cone, is the wall of the cone uniformly thick? How thick?
How wide is the base?
Is the base a hollow stem? Or is it a solid cylinder made of concrete?
If it is hollow, how thick is the wall of the base cylinder?
Depending on design, it could be a complicated problem.
There is also the issue of how realistic you have to be.
Do you need to buy a whole number of liters of paint, or can you buy exactly the amount you need, even if it is a crazy decimal number, such as 0.17 liter?
What about the cement? Can you order 0.43 m^3?
The simplest, if ugly, design would be a cylinder with a conical hollow at the top.
In cross section, it would look like this:
VOLUME:
The volume of that is the volume of the cylinder minus the volume of the cone.
If the cylinder and cone were 1 m in diameter (0.5m radius), with a cylinder height of 1m, and a cone height (depth) of 0.25m, the volume, in m^3 would be
At $100/m^3, the cement to make that would cost $52.36.
SURFACE AREAS:
The surface to be painted would be the lateral surface of the cylinder,
calculates as  ,
plus the lateral surface of the cone,
calculated as 
For the measurements assumed above, the area in m^2 would be
COST:
The cost for 1 liter of paint and 0.53m^3 of concrete would be
$17.50 + $53.00 = $70.50
The fountain I would design, in cross-section, would look like this:
 The height of the cone is half the diameter.
VOLUMES:
The volume of cement in the cone is the volume of a cone with a 0.5m radius and a 0.25m height, minus the volume of a cone with a 0.45m radius and a 0.225m height. In  , it is
The volume of cement in the base is the volume of a cylinder with a 0.1m radius and a 1m height, minus the volume of a cone with a 0.08m radius and a 1m height. In , it is
The total volume of concrete is
 .
At $100/m^3 that would cost $2.91.
SURFACE AREAS:
The outside cone has a radius of 0.5m and a height of 0.25m.
Its slant height, in m, is
The perimeter of the base, in m, is  , because the diameter is 1m.
The outside surface area of the cone, in  , is
 .
The surface area of the cone insides plus the rim is about the same.
The outside surface area of the cylindrical base, in , is
The total area to be painted, in , is about
That would require less than 1 liter of paint.
COST:
Buying 1 liter of paint, plus 0.03m^3 concrete would cost
$17.50 + $3.00 = $20.50
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