I'm really struggling with the following question:

A river running due east has straight parallel banks. A vertical post stands with its base, P, on the north side of the river. On the south bank are two surveyors, A who is to the east and B who is to the west of the post. A & B are at a distance 2/7a apart and the angle APB = 150 degrees. The angles of elevation from A and B of the top Q, of the post are 45 degrees and 30 degrees. Find in terms of a, the width of the river and the height of the post.

what I have figured out so far is that:

QP = AP (tan45 = 1)

PB = APsqrt3 (half equilateral)

BQ = 2AP (half equilateral)

AB is the longest side of the triangle ABP (angle APB = 150 degrees)

after struggling for 30mins I checked the answers at the back of the book and they are given as

a sqrt3/49 and 2a/7 sqrt7...certain things look familiar here but...

my problems trying to find just the width are as follows:

- if I was to assume that a is the length of either side opposite angle A in triangles ABP or APQ then this would be impossible because AB is 2/7a but is the longest side (opposite P which is 150 degrees). is a just an arbitrary value that AB is 2/7 of?

- If I create a line PR that is perpendicular to AB, PR would be the width of the river but how using pythagoras theorem, the sine, or cosine theorems can I make an equation with respect to a without knowing the relationship algebraically between AB and any other lines, the lengths of AR or BR, or any of the angles? (don't know the angles RPA or RPB)

- If I assume that 2/7a is a typo and that it should read 7/2a and decide that a is now the length of AP or BP this too would be impossible because

a + a sqrt3 is less than 7/2a and thus you can't make a triangle.

Is there something important I am missing, or are my assumptions terribly flawed?...please help

A river running due east has straight parallel banks. A vertical post stands with its base, P, on the north side of the river. On the south bank are two surveyors, A who is to the east and B who is to the west of the post. A & B are at a distance 2/7a apart and the angle APB = 150 degrees. The angles of elevation from A and B of the top Q, of the post are 45 degrees and 30 degrees. Find in terms of a, the width of the river and the height of the post.

what I have figured out so far is that:

QP = AP (tan45 = 1)

PB = APsqrt3 (half equilateral)

BQ = 2AP (half equilateral)

AB is the longest side of the triangle ABP (angle APB = 150 degrees)

after struggling for 30mins I checked the answers at the back of the book and they are given as

a sqrt3/49 and 2a/7 sqrt7...certain things look familiar here but...

my problems trying to find just the width are as follows:

- if I was to assume that a is the length of either side opposite angle A in triangles ABP or APQ then this would be impossible because AB is 2/7a but is the longest side (opposite P which is 150 degrees). is a just an arbitrary value that AB is 2/7 of?

- If I create a line PR that is perpendicular to AB, PR would be the width of the river but how using pythagoras theorem, the sine, or cosine theorems can I make an equation with respect to a without knowing the relationship algebraically between AB and any other lines, the lengths of AR or BR, or any of the angles? (don't know the angles RPA or RPB)

- If I assume that 2/7a is a typo and that it should read 7/2a and decide that a is now the length of AP or BP this too would be impossible because

a + a sqrt3 is less than 7/2a and thus you can't make a triangle.

Is there something important I am missing, or are my assumptions terribly flawed?...please help

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