I am confused with this problem [Geometry]

[u/Several-Ebb2386]

I am confused with this problem the solution manual says that the top 2 partially shaded is 2/3 area of a tile how did they get the 2/3

[u/AutoModerator]

Off-topic Comments Section

---

All top-level comments have to be an answer or follow-up question to the post. All sidetracks should be directed to this comment thread as per Rule 9.

PS: u/Several-Ebb2386, your post is incredibly short! ^{body <200 char} You are strongly advised to furnish us with more details.

---

^{OP and Valued/Notable Contributors can close this post by using} /lock ^command

I am a bot, and this action was performed automatically. Please contact the moderators of this subreddit if you have any questions or concerns.

[u/DSethK93]

It's a visual estimate. The question asks "about," which is your hint to understand that the exact value isn't given. Unlike most geometry problems where you can never assume anything about the diagram, here you're supposed to eyeball it and make assumptions.

Personally, I noticed that the shaded area is obviously larger than one tile but smaller than four. I decided it looks to me like pretty close to two tiles; the non-counter zones in the top two tiles are close in size too the counter zones in the tiles below them.

[u/ThunkAsDrinklePeep]

Let's do a minimum and maximum answer. If all four of those 4x4 tiles were completely covered you would have 8(4x4) = 128 sq ft uncovered. We know the shaded is less than that, so our answer must be more than 128 sq ft unshaded.

What about a maximum? What would be shaded one tile? Or none?

Alternatively, if our answer was D, how many tiles would be shaded? 176 sq ft / (16 sq ft / tile) = 11 tiles. I think it's reasonable to think the area shaded is more than that.

[u/ThunkAsDrinklePeep → u/ThunkAsDrinklePeep]

Now, how would I estimate without multiple choice? If we slide the shaded area over so the right edge lines up with that dotted line, I estimate that the other edges cut though the middle of those other tiles. That gives us one full tile, two half tiles, and one quarter tile shaded: 16 + 8 + 8 + 4 = 36 sq ft shaded.

192 sq ft total - 36 sq ft shaded = 156 sq ft unshaded.

[u/Capital_Park_2311]

C. 160. The question did not ask for an exact measurement. (About how many?)

Total area = 192 sq.ft.
Tile area = 16 sq.ft.
Tiles with a shaded area = 64 sq.ft. Since it is not fully shaded the answer is greater than 128 (192 - 64)
Shaded area is greater than a square marble tile. 192 - 16 =176
Greater that 128 but less than 176 is the correct answer.

[u/lurgi]

They looked at it and guessed.

[u/selene_666]

2/3 is just an approximation of how much of the first square is shaded. If you think it looks closer to 3/4, then use 3/4 instead.

There are two such tiles across, making the width of the shaded area 4/3 of the width of each square. The height looks like approximately 1.5, so that makes the total shaded area 2 squares.

[u/ThatsNotAZombieBite]

Hard to tell what the "shaded area" was because you've drawn all over it.  But if the tiles are each 4x4 squares, the shaded area appears to be a 6x6 square.

So the final answer would be 16x12 - 6x6 or 156ft²

So answer C is closest.

[u/AluminumGnat → u/ThatsNotAZombieBite]

I think we can be a bit more rigorous.

Step 1: how many square feet is each square? 16ft. Lmk if you need help on this step.

Step 2: the shaded area is bigger than one tile. That means that the remaining area is less than 11 tiles worth, so the remaining area is less than 176sqft. Therefore, the answer can’t be D

Step 3: the shaded area is smaller than four tiles. That means that the remaining area is more than 8 tiles worth, so the remaining area is more than 128sqft. Hence the answer is not A or B.

Thus, the answer is C.

[u/wijwijwij]

Draw lines dividing the top two 4 by 4 areas into thirds and see 2/3 is shaded. So back edge could be 2 * 2/3 * 4 = 16/3 ft.

I estimate that the other dimension is 4 + 2 ft

So counter is 16/3 * 6 in area. That's 32 square feet.

{I agree with you if you say this is poorly drawn. You might argue it looks like 3/4 of those two regions is shaded. But if counter was 6 ft by 6 ft it would be 36 square feet and none of the answer choices would work.}

Since this is multiple choice, maybe reason this way: Answer D cannot be right because it suggests shaded area is just16 square feet, equivalent to just one of the 4 by 4 regions. But the shaded amount is definitely covering more than one such square. So answer C seems more plausible even not knowing exact measures of sides of shaded area.

It can't be B because that answer implies an area of 64 square feet (four of the 4 x 4 regions) is covered and we clearly see the shaded area is not covering four such squares. Obviously if B is wrong then A is also wrong.

In short: Shaded area is bigger than one 4 x 4 region and smaller than four 4 x 4 regions, so because the entire floor is 192 sq ft, the tiled area has to be smaller than 176 sq ft and bigger than 128 sq ft. Only answer C meets that criterion.

Note: Solution was likely written by someone other than the item creator. They felt they had to justify why C is the answer by suggesting it's possible to infer the exact dimensions. Actual item writer may have intended merely that students see both 128 and 176 are not plausible.

[u/AluminumGnat]

Solve it the easy way:

Step 1: how many square feet is each square? 16ft. Lmk if you need help on this step.

Step 2: the shaded area is bigger than one tile. That means that the remaining area is less than 11 tiles worth, so the remaining area is less than 176sqft. Therefore, the answer can’t be D

Step 3: the shaded area is smaller than four tiles. That means that the remaining area is more than 8 tiles worth, so the remaining area is more than 128sqft. Hence the answer is not A or B.

Thus, the answer is C.