12 Days of Christmas GMAT Competition 2025 - Day 9: A point light : Two-part Analysis (TPA)

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25 Dec 2025, 10:18

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A point light source shines equally in all directions.
The light intensity measured at a sensor is inversely proportional to the square of the distance from the source.
When the distance is 6 meters, the intensity is 8 units.

Intensity(I) = k * 1/d^2 ; where d = distance from the source in meters

8 = k/6^2 ; k = 8*6^2 = 288

I = 288/d^2

d = 2; I = 288/2^2 = 72; Not an option
d = 4; I = 288/4^2 = 18; Not an option
d = 12; I = 288/12^2 = 2; Possible

Distance	12
Intensity	2

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25 Dec 2025, 10:52

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Bunuel

according to info
k = intneisty * distance * distance
k = 8 * 6 * 6 = 288
for distance = 12, intensty = 288/ (12*12)
= 2

answer is 12, 2

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25 Dec 2025, 10:54

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Bunuel

I = k/D^2

8 * 36 = k

k = 288

So we have to find a value for which the k comes out to be 288

Let say intensity =2

D^2 = 288/2 = 144

D = 12

We do have this combination, hence we don't have to test out any other values.

Distance = 12
Intensity = 2

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25 Dec 2025, 11:01

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intensity = k / (d)^2
8=k / 36
k= 288

now we can try value of intensity with given value of k and find distance

when i=2, d= 12

Bunuel

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25 Dec 2025, 11:12

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Bunuel

Light intensity is INVERSELY proportional to square of the distance.

8 is inversely proportional to 36

8 = k(1/36)

K = 8*36

K= 4*2*9*4

Light intensity = (4*2*9*4) * (1/d^2)

If d =2, then light intensity = 4*2*9 = 72 ( not in the options)

If d = 4, then light intensity = 2*9 = 18 ( not in the options).

If d = 12, then light intensity = 2. ( Matches with the options).

d=12 , LI = 2

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25 Dec 2025, 11:16

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Let Intensity = K/distance^2 hence 8 = K/36 -------> K = 8*36 = 288

From options for distance 12 Intensity = 288/12^2 = 2

Bunuel

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25 Dec 2025, 11:32

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Intensity (I) is inversely proportional to the square of the distance (d)
I = x / d^2

Find x: 8 = x/36 -> x = 8*36 = 288

Check answer choices considering that as d increases, I decreases.

The only possible answers are distance = 12, intensity = 2
2 = 288/12^2 = 288/144

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25 Dec 2025, 11:35

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I= K/(d^2)
8=K/(6^2)
K=8*36

I = (8*36)/d^2

if d=12, I =2 satisfy.

Ans 12 & 2

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25 Dec 2025, 11:42

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I inversely proportional to square of distance

$I1 * D1^{2} = I2 * D2^{2}$

$8 * 6^{2} = I2 * D2^{2}$

$2*2^{2} * 6^{2} = I2 * D2^{2}$

$2 * 12^{2} = I2 * D2^{2}$

Distance = 12, Intensity = 2

Ans: 12, 2 or C, A

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25 Dec 2025, 11:53

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Bunuel

Intensity I varies as [1][/d^2], so I = [k][/d^2]

Given d = 6, I = 8
8 = k/36
k = 288

so I = 288/d^2

Lets pick a distance from the left column that makes I match one of {2,4,12,16,24,32}

Lets take 12
I = 288/12^2 = 288/144 = 2

Distance = 12
Intensity = 2

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25 Dec 2025, 11:53

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let intensity and distance be represented by I & D respectively.
given I is inversely proportional to the square of the D from the source.
I=k/(D^2) or k=I*D^2 k-constant
k=8*6^2=8*36
to find another set of the I and D we just need to rearrange above calculated value of k so that its the product of a perfect square denoting distance and an integer denoting I.
k=8*36=9*4*4*2=12^2*2
I=2 and D=12 are the correct choices

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25 Dec 2025, 12:53

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Bunuel

If intensity is inversely proportional to distance squared, that means:

$intensity=\frac{k}{distance^2}$, where k is a constant.

Subbing in the example we have, we get:
$8=\frac{k}{6^2}$ which results in $k=288$.

So now we're looking for:
$intensity=\frac{288}{distance^2}$ which can be rearranged into $intensity*distance^2=288$.

The prime factorization of $288=2^5*3^2$

Looking at the answer choices, only 12 and 24 would contribute a 3, so they are most likely to be the value for distance, because we need a $3^2$.

$24^2=3^2*2^6$. 24 gives too many factors for 2, so distance has to be 12.

$12^2=3^2*2^4$, so we need one more 2 - thus intensity needs to be 2.

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25 Dec 2025, 14:07

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The inverse relationship can be expressed as Intensity (I)=1/distance squared (d^2)x Constant (k)
Given I=8 and d=6, we can find k
8=1/6^2xk
k=288
So intensity=288/d^2. From here we can test cases using the answer choices starting with intensity of 2
2=288/d^2
d^2=288/2=144
d=12
Given we have both 2 and 12 among the answer choices, there is no need for further testing so
Intensity = 2
Distance= 12

Bunuel

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25 Dec 2025, 14:16

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Bunuel