Concepts addressed: **percentages**, **unit conversion**, **cylinder volume**, **cylinder area**, **equilateral triangle area**, **regular hexagon area**.

Recommended grade: **7th**.

Difficulty level: **Basic**.

A basic version of Building a swimming pool is also available.

**Scenario:**

This is the year – **you’re 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.

- You’ve decided the pool should be
**circular**, with**a radius of six feet**and**as deep as you are tall**. How much water will you need if you want to fill it up**nine-tenths of the way**(so that it doesn’t spill)?

- You want to tile the entirety of the inside with regular hexagonal tiles that have a side 3 in long. How many of those will you need?
**Better buy 5% more**in case some of them break in the process.

**A pack of thirty 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:**

- Volume calculator – https://www.omnicalculator.com/math/volume
- Cylinder volume calculator – https://www.omnicalculator.com/math/cylinder-volume
- Circle calculator – https://www.omnicalculator.com/math/circle
- Percentage calculator – https://www.omnicalculator.com/math/percentage
- Hexagon calculator – https://www.omnicalculator.com/math/hexagon
- Conversion calculator – https://www.omnicalculator.com/conversion/conversion-calculator

**Question 1 hints:**

**Question 2 hints:**

**Question 3 hints:**

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

**Step-by-step solution:**

**circular**, so the shape of it (and therefore the water inside too) will be

**cylindrical**. From the scenario, we know that

**the radius of the base**, r, is 6 ft. Also, if we are 6-feet tall, then this will be the pool’s depth, so h_cylinder = 6 ft. However,

**we don’t want to fill it all the way**, but only nine-tenths of the way. Therefore,

**the height the water, h,**will be:

*h = h_cylinder * 9/10 = 6 ft * 9/10 = 5.4 ft*.

This gives us all the information we need to calculate the volume with the cylinder volume calculator:*V = π * r ² * h = π * (6 ft)² * 5.4 ft ≈ 610.4 ft³*.

**the bottom and the lateral face of our cylinder**. We can think of the latter as

**a rectangle**(whose two sides have been glued together) with one side equal to the cylinder height, h = 6 ft, and the other equal to

**the perimeter of the circle**that is the base, b. We can use the circle calculator to find b:

*b = 2 * π * r = 12π ft*.

Now we can use that same calculator and get

**the area we want to tile**:

*A = base_area + lateral_area = π * r*.

*+ b * h =**²**π** (6 ft)*+ 12**²**π*ft * 6 ft*≈*339.12 ft²Now that we know how much area we’re working with, we need to get the number of tiles required to tile it. A single tile is** a regular hexagon** with side a = 3 in. For our calculations, **we need the units to agree**, so let’s use the conversion calculator to turn a into feet: a = 3 in = 0.25 ft.

Recall that a regular hexagon consists of **six equilateral triangles**, with sides equal to the hexagon’s side. Also, there is a nice formula for the area of an equilateral triangle that needs only the length of its side. In particular, this means that we are able to calculate **the area of a single tile** with the hexagon calculator:*A_tile = 6 * area_triangle = 6 * (a² * √3) / 4 = 6 * ((0.25 ft)² * √3) / 4 ≈ 0.16 ft²*.

Therefore, we’ll need:

*A / A_tile*2,119.5

*339.12 ft**≈**²*/ 0.16*ft*=*²*tiles to cover the whole area. Clearly, you can’t buy half of a tile, so the actual number we need is 2,120.

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 needed, then this means that we need 100% + 5% = 105% of what we calculated above. According to the percentage calculator this is 105% * 2,120 = 2,226 tiles.

**the cost of the tiles**. From Question 2 we know that we need 2,226 of them. Now we must see how many packs of 30 tiles this translates to:

*2,226 / 30 = 74.2*.

Therefore, we’ll need **75 packs** (since we cannot just buy 0.2 of a pack). This, in turn, boils down to*cost = 75 * pack_cost = 75 * $15 = $1,125*.

Now let’s subtract from that the amount you’ve already saved to get **how much we still need**:*money_needed = cost – money_saved = $1,125 – $80 = $1,045*.

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 1,045 / 50 = 20.9, or rather 21 weeks, to get that much money. But summer will be over in 21 weeks!

Oh well, maybe** next year will be the year**?

A bonus:what are you teaching next week? We'd love to prepare a scenario for you 🙂