One of the hardest ACT questions ever

This is said to be one of the hardest ACT questions. #math #maths #mathematics #shorts

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[MindYourDecisions]

Extra credit: what is the exact area to 3 decimal places? (Edit: added to 3 decimal places)

[geoninja]

I paused the video before the choices appeared, so had no idea of just how 'approximate' I needed to

[leickrobinson]

Another way to show this rigorously, but still, without a calculator, is to show the bounds of the area.

It has to be inside the 16x14 rectangle, so the area is less than 224.

By symmetry, we can see that the point (12, 12) is also on the circle, so we can also construct a shape wholly within the blue area by the rectangle {(0, 0), (0, 12), (12, 12), (12, 0)}, the triangle {(0, 12), (12, 12), (6, 14)}, and triangle {(12, 12), (12, 0), (16, 4)}. The total area of this shape is 144+12+24 = 180. So, the blue area must be greater than 180.

Hence, answer C.

[mstmar]

i did it completely differently. we know the wanted area is less than 14*16 = 224, since that rectangle includes the whole shape. for a lower bound, we need 1 more point, the other point at y = 12. we know that the x distance to the center of the circle will be the same, so we just need to add an extra 6-0 to the center's x. so (12, 12) is another point on the circle. we can then create a square (12*12) and 2 triangles (on top, base 12, height 2, and on the right, base 12 and height 4) using the points on our circle. lower bound is 180. so our area is between 180 and 224, so C

[spiderjerusalem]

my first glance was to use integration with polar coordinates.

(x–6)²+(y–4)² = 100
x²+y²–12x–8y+36+16=100
r²–2(6cosψ+4sinψ)r = 48
r = 6cosψ+4sinψ + √(48+(6cosψ+4sinψ)²)

let a = 6cosψ+4sinψ

r² = 2a² + 48 + 2a√(48+a²)

since:
a² = 36cos²ψ+16sin²ψ+24sin2ψ
= 20cos²ψ+16+24sin2ψ
= 10cos2ψ+6+24sin2ψ

thus:
½r² = 10cos2ψ+24sin2ψ+6
+ a√(48+a²)

area = ∫ ½r² dψ [0, ½π]
= 24+3π + ∫ a√(48+a²) dψ [0, ½π]
= 24+3π + dψ [0, ½π]

and the rest shall be left for readers to

[toomanyhobbies]

Use a rectangle as an overestimate: 14 x 16 = 224, which is above the range, so less than 225. Option 'C'.

[Waldlaeufer]

I would calculate 16 × 12 = 192 to get the area of the rectangle.

[fsyi]

it will be more interesting to compute the area difference between the sum of 1st and 3rd quadrants and the sum of 2nd and 4th quadrants

[emiltonklinga]

Yeah, I do agree with those who say that the only rigorous way to do this is to both overestimate and underestimate the area. Just because your approximation lands in the bounds of 175 - 225 doesn't mean it is the correct answer. I mean, if I wanted to I could do an approximation that was under 175 but that wouldn't give me the correct answer.

We can overestimate the area by doing the same calculation as you but using pi is approx 3.2. That gives us a total area of 204 and the actual area has to be smaller than that.

Then we can underestimate by using pi is approx 3, a rectangle with sides 6 and 12 and finally a trapezoid with sides 6 and 10 and height 4. That gives us a total area of 179 and the actual area has to be larger than that.

Hence, 179 < A < 204 which implies 175 < A < 225.

[yorkcheung]

Wow videos getting much quicker nowadays

[ddichny]

I did the 12x16 rectangle as an approximation, giving 192. The chord is misses on the top eyeballs about the same area as the corners that exceed the shaded area on the right.

[dashingclasher]

that was a very clever use of approximation, i need to brush up my geometry

[ivankaznacheyeu]

one should find both lower and upper boundary

[deankremer]

Given the fairly wide multiple-choice ranges here, I'd simply calculate that the circle area is 314 and note that we clearly have around 2/3 of the circle. That's 200-ish, which is plenty close enough here. Would I do a quick check of the quarter circle (about 78) + the two rectangles 64 (bottom) + 48 (left)? Sure, and I'd know I was coming in a little low. Precision is the enemy of speed under these circumstances.

[GD-mwkd]

Once the one quarter of the circle was visualised, I immediately dived to think in terms of quarters out of remaining areas

[Kevinjohn]

Can we get a video and how the points are plotted👍

[winterfeuer]

The blue area is approximately 2/3 of thr full circle. That easyly is 100*π
So I also come to roundabout 200 square units.

Sorry 4 bad english :(

[reecec]

I just did 12 x 16, which is somewhat equivalent to 14 x 14. Which is 196. So it was in the right bracket for answer C.

[antonyjoseph]

Sometimes it's these easy questions that get us

[SimonLanghof]

I did it almost the same way but I went for 13x6 for the left rectangle and used 3.14 for pi.