A Maths Applied student holidaying in Sydney notices that the Centrepoint tower has an angle of elevation of 15deg20'. After walking 400m directly towards the tower she now observes the elevation to be 24deg33'

I assume you are talking about angles of elevation, and want to find the height of the tower.

If she ends up x meters from the tower, then

h/x = sin 24°33'
h/(x+400) = sin 15°20'

equating the values for x,

h/sin 24°33' = h/sin 15°20' - 400
h/.415 = h/.264 - 400
h = 290.23

I assume you are talking about angles of elevation, and want to find the height of the tower.

If she ends up x meters from the tower, then

h/x = tan 24°33'
h/(x+400) = tan 15°20'

equating the values for x,

h/tan 24°33' = h/tan 15°20' - 400
h/.45678 = h/.27419 - 400
h = 274.37

How would I find out how far away from the base of the tower she was when she made her first measurement of 15deg20'

To solve this problem, we can use trigonometry and the concept of similar triangles.

Let's consider a right-angled triangle with the Centrepoint tower as the vertical side, and the distance between the observer's initial position and the base of the tower as the base. The angle of elevation observed initially is 15 degrees 20 minutes, which we'll convert to decimal degrees for simplicity. Now, we can label the sides of this triangle as follows:

- The vertical side (Centrepoint tower) = h
- The base (initial distance from observer to tower) = b

Using trigonometry, we can determine the height of the Centrepoint tower (h) by forming a tangent ratio:

tan(angle of elevation) = opposite/adjacent
tan(15.3333 degrees) = h/b

Now, let's move to the second observation point after the observer walked 400m towards the tower. At this new location, the angle of elevation is 24 degrees 33 minutes. Again, we'll convert this angle to decimal degrees.

This time, we can consider a new triangle formed by the observer's new position, the top of the Centrepoint tower, and a point directly below the tower at the same distance from the observer as before (400m). Let's label the sides of this triangle as follows:

- The vertical side (Centrepoint tower) = h
- The base (new distance from observer to tower) = b - 400m

Using the same tangent ratio as before:

tan(angle of elevation) = h/(b - 400)
tan(24.55 degrees) = h/(b - 400)

We now have a system of equations with two unknowns (h and b) but we can simplify it by setting the two expressions for h equal to each other:

tan(15.3333 degrees) = h/b
tan(24.55 degrees) = h/(b - 400)

By substituting the first equation into the second equation, we get:

tan(24.55 degrees) = (h/b)/(1 - 400/b)
tan(24.55 degrees) = (tan(15.3333 degrees))/(1 - 400/b)

Now, we can solve this equation to find the value of 'b', which represents the distance from the observer's initial position to the base of the Centrepoint tower. Once we know 'b', we can substitute it back into the first equation to find 'h', which represents the height of the tower.