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Light Height is a puzzle in Professor Layton and the Last Specter.

## Puzzle

US Version

A single light hangs from a beam above the floor of this dilapidated warehouse.

There are two posts, both of which are a foot tall, separated by a distance of 15 feet. The light casts a 3 foot long shadow from the left post and a 2 foot long shadow from the right post.

Figure out how high the light is hanging above the floor.

UK Version

A single light hangs from a high-up beam in this dilapidated warehouse. Down on the floor, there are two posts 1 m high and 15 m apart. The light casts a 3 m long shadow from the left post and a 2 m long shadow from the right.

How high, in metres, is the light from the floor?

## Hints

Click a Tab to reveal the Hint.

US Version

Choose one of the posts, and visualize a large triangle formed between the light, the spot on the floor directly under the light, and the farthest point of the post's shadow. Then imagine a smaller triangle formed between the top of the post, the bottom of the post, and the tip of the shadow.

These two triangles are mathematically similar. In other words, the larger one is proportional to the smaller one.

UK Version

Choose one of the posts and visualise a large triangle formed between the light, the spot on the floor directly under the light and the furthest point of the post's shadow.

These two triangles are mathematically similar. In other words, the larger one is proportional to the smaller one.

US Version

Let's say that the height of the light equals X, and the distance on the floor between the light and the post on the left equals Y.

The shadow cast by the post on the left extends 3 feet.

X:(Y:3)=1:3, therefore 3X=Y+3

UK Version

Call the height of the light X. Call the distance on the floor between the light and the post on the left Y. The shadow cast by the post on the left extends 3 m.

X : (Y + 3) = 1 : 3 and therefore 3X = Y + 3

US Version

The shadow of the post on the right is 2 feet long.

X:(15-Y+2)=1:2, therefore 2X=17-Y

UK Version

The shadow of the post on the right is 2 m long.

X : (15 - Y + 2) = 1 : 2 and therefore 2X = 17 - Y

US Version

By combining the solutions from the equations of Hint 2 and Hint 3, you get:

5X=20, therefore X=4

The height from the light to the floor equals X.

UK Version

By combining the equations from Hint 2 and Hint 3 and solving them simultaneously you get:

5X = 20 and therefore
X = 4

X is the height of the light, so it looks like you've found the answer.

## Solution

### Incorrect

US Version

UK Version

Check your calculations and try again!

### Correct

US Version

That's right! The light is 4 feet tall!

As shown in the diagram, make two triangles, ABC and EBF, on the left of the light and two more, ADC and GDH, on the right.

AC:BC = EF:BF therefore
X:(Y+3) = 1:3
AC:CD = GH:HD therefore
X:(15-Y+2) = 1:2

Solving both of these equations will give you the value of X.

UK Version

That's right! The light is 4 m up.

As shown in the diagram, make two triangles, ABC and EBF on the left of the light and two more, ADC and GDH, on the right.

AC : BC = EF : BF therefore
X : (Y + 3) = 1 : 3
AC : CD = GH : HD therefore
X : (15 - Y + 2) = 1 : 2

Using both these equations, you can work out the value of X.

A big thanks to http://professorlayton4walkthrough.blogspot.com

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