A homeowner is deciding on the size of tiles to use to fully...

A homeowner is deciding on the size of tiles to use to fully tile a rectangular wall in her bathroom that is 80 inches by 40 inches. The tiles are squares and come in three side lengths: 8 inches, 4 inches, and 2 inches. State if you agree with each statement about the tiles. Explain your reasoning. a. Regardless of the size she chooses, she will need the same number of tiles. b. Regardless of the size she chooses, the area of the wall that is being tiled is the same. c. She will need two 2 -inch tiles to cover the same area as one 4 -inch tile. d. She will need four 4 -inch tiles to cover the same area as one 8 -inch tile. e. If she chooses the 8-inch tiles, she will need a quarter as many tiles as she would with 2inch tiles.

A homeowner is deciding on the size of tiles to use to fully tile a rectangular wall in her bathroom that is 80 inches by 40 inches. The tiles are squares and come in three side lengths: 8 inches, 4 inches, and 2 inches. State if you agree with each statement about the tiles. Explain your reasoning.
a. Regardless of the size she chooses, she will need the same number of tiles.
b. Regardless of the size she chooses, the area of the wall that is being tiled is the same.
c. She will need two 2 -inch tiles to cover the same area as one 4 -inch tile.
d. She will need four 4 -inch tiles to cover the same area as one 8 -inch tile.
e. If she chooses the 8-inch tiles, she will need a quarter as many tiles as she would with 2inch tiles.

Key Concepts

Tile Count and Area Relationship

The number of tiles required to cover a given area is determined by dividing the total area by the area of a single tile. Thus, smaller tiles cover less area per unit, and more tiles are needed to achieve the same overall coverage, illustrating an inverse relationship between tile area and tile count.

Scaling of Area with Side Length

When the side length of a square is scaled by a factor, its area scales by the square of that factor. For example, if the side length is doubled, the area increases by a factor of four. This concept is key in understanding how the number of tiles needed changes when opting for different tile sizes.

Square Area Formula

For a square, the area is given by squaring the length of its side. This formula, A = side², is crucial in comparing different tile sizes, as it allows one to compute the area of any square tile, regardless of its specific side length.

Area Calculation

This concept involves determining the area of the space to be tiled, in this case by multiplying the length and width of the wall. It provides the total surface area that must be covered by the tiles, and it is invariant regardless of the size of the tiles used, ensuring that different tiling choices all cover the same total area.

Transcript

00:01 So in this problem, we're told that a homeowner is going to tile one wall in their bathroom.

00:06 And we know for the wall that it's 80 inches by 40 inches, meaning we could find the area for that wall by multiplying them together.

00:13 Now, the different pieces of tile, they're all squares.

00:17 So one of the squares is 8 inches, meaning that all the sides are 8 inches.

00:21 One is 4 inches, so that would be a little bit smaller.

00:24 And the last one's 2 inches, so that's even smaller.

00:28 Okay, so what we're going to do is we're going to analyze.

00:30 Each of these statements.

00:32 So let's first look at part a.

00:34 Part a says regardless of the size she chooses, she will need the same number of tiles.

00:39 Well, let's think about it.

00:41 If we are using larger tiles, we won't have to use as many as if we are using the smaller tiles.

00:46 So a is going to be false.

00:49 You're not going to use the same number of tiles to cover the space because they're all different sizes.

00:55 Okay.

00:56 Next, let's look at part b.

00:58 It says regardless of the size she chooses, the area of the wall that is being tiled is the same.

01:04 Well, no matter what, we're still covering the same wall.

01:07 So yes, this is going to be true.

01:08 The area of the wall will not change depending on how many tiles we use.

01:13 All right.

01:14 Next, let's look at part c.

01:16 She said she will need two two -inch tiles to cover the same area as one four -inch tile.

01:21 Well, if i took two -inch tiles, so this would be two inches and this would be two inches.

01:26 Let's find the area.

01:28 Well, this length is two.

01:29 This would be two and two, which would have a length of four.

01:33 So the area of these would equal to two times four, which is eight.

01:38 And it says it would be the same as if we were to take one of the four -inch tiles.

01:44 So if i took one four -inch tile, the length would be four, and the width would be four.

01:49 So to find the area, we would do two, or excuse me, we would do four times four, which is 16.

01:54 Well, eight and sixteen aren't the same number.

01:56 Therefore, the answer to the part c is false.

02:00 All right.

02:01 Next, let's take a look at part d...

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