# 3D geometry – Building a swimming pool (basic)

Concepts addressed: percentages, rectangular prism volume, rectangular prism area, unit conversion, square area.
Recommended grade: 7th.
Difficulty level: Basic.
An advanced version of Building a swimming pool is also available.

Scenario:

This is the year – you’re finally building an in-ground swimming pool in your backyard! You can already see all the envious faces, but first someone’s got to do the dirty work.

1. You’ve decided the pool should be rectangular, with dimension eight by thirteen feet and as deep as you are tall. How much water will you need for it?

1. You want to tile the inside of the pool with square eight by eight-inch tiles. How many of those will you need? Better buy 5% more in case some of them break in the process.

1. A pack of ten tiles costs \$15, and you get \$50 allowance every week. How long will it take to save for them if you already have \$80 saved, and you’ll be putting all your allowance towards it?

Useful calculators:

Question 1 hints:

Hint 1
What shape does the pool have in three dimensions? Can you describe its sides?
Hint 2
How do we calculate the volume of a parallelepiped (rectangular prism)?

Question 2 hints:

Hint 1
How do we calculate the area we will have to tile? Remember that we don’t tile the top of our solid.
Hint 2
What is the area of a single tile? If we know how much area we have to tile, how many tiles will cover it all?
Hint 3
Remember that we need to have all values expressed in compatible units.
Hint 3
If we know how many tiles cover the area, how can we compute the additional 5%?

Question 3 hints:

Hint 1
How much money will be needed to buy all the necessary tiles? How much do you still have to save?
Hint 2
Knowing how much money we need, and that you get \$50 allowance a week, how can we calculate the number of weeks that has to pass?

Solutions (WARNING: depend on the height, example for height being 6 feet):

Question 1
V = 624 ft³
Question 2
tiles = 842
Question 3
weeks = 24

Step-by-step solution:

Question 1
We want the pool to be rectangular, so the shape of it (and therefore the shape of the water inside) will be a rectangular prism. From the scenario, we know that its sides, a & b, are 8 ft and 13 ft respectively. Also, if we are 6-feet tall, the pool will also be this deep, and therefore the height of the prism, h, is 6 ft.

This gives us all the information we need to calculate the volume with the rectangular prism calculator:
V = a * b * h = 8 ft * 13 ft * 6 ft = 624 ft³.

Question 2
The tiles will cover the bottom and the four sides of the pool, so we need to calculate the area of the base and the lateral faces of our rectangular prism. All of those are rectangles, the former with sides a = 8 ft and b = 13 ft, and there are two kinds of the latter, each appearing twice (on opposite sides). The first kind is above the 8-foot side of the base, and therefore has sides a = 8 ft and h = 6 ft. The second is above the 13-foot one, having sides b = 13 ft and h = 6 ft. This allows us to calculate the area we want to tile:
A = base_area + lateral_area = a * b + 2 * a * h + 2 * b * h = 8 ft * 13 ft + 2 * 8 ft * 6 ft + 2 * 13 ft * 6 ft = 356 ft².

Now that we know how much area we’re working with, we need to find the number of tiles that will tile it. A single tile is a square of side 8 in. For our calculations, we need the units to agree, so let’s use the conversion calculator to write the tile side in feet: a_tile = 8 in = 2/3 ft. Therefore, the area of a square calculator states that a single tile has an area of:
A_tile = (a_tile)² = (2/3 ft)² = 4/9 ft².
Therefore, we’ll need:
A / A_tile = 356 ft² / (4/9 ft²) = 801
tiles to cover the whole area.

Lastly, we need to add to that number the additional 5% in case some of the tiles break. If we want to buy 5% more than what is required, then this means that we need 100% + 5% = 105% of what we calculated above. According to the percentage calculator, this is 105% * 801 = 841.05, which tells us that we need to buy 842 tiles.

NOTE: If you’ve rounded the area of a single tile to 0.44 ft² instead of keeping it as a fraction, then you could have gotten a slightly different result. You can see that in the above calculations, there is no approximation anywhere, which from the point of view of real-life scenarios, is a great thing. However, in many cases, rounding numbers is very useful. As a general tip to avoid large discrepancy we suggest doing them as late as possible.

Question 3
Let’s begin by calculating the cost of the tiles. From Question 2, we know that we need 842 of them, which translates to 85 packs of 10 (since the tiles are not sold separately). This boils down to
cost = 85 * pack_cost = 85 * \$15 = \$1,275.

Now let’s subtract from that the amount you’ve already saved to get how much we still need:
money_needed = cost – money_saved = \$1,275 – \$80 = \$1,195.

Lastly, we must see how many allowances (and therefore weeks) it will take to save such a sum. Since we get \$50 every week, we need 1,195 / 50 = 23.9, or rather 24 weeks to get that much money. But summer will be over in 24 weeks!

Oh well, maybe next year will be the year?

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