Bobby is standing near a lighthouse. He measured the angle formed from where he stood to the top of the lighthouse and it was 30 degrees. Then he backed up 40 feet and measured the angle again and it was 25 degrees. Solve the height of the lighthouse.

Or, method two simplifies a bit if you're not afraid of the cot(x) function:

h cot25 - h cot30 = 40

I also got the answer of 96.98 ft, but the answer key says 18.4 ft. Who is right?

If the answer key says 18.4 ft, it's possible that they made a mistake or gave you a different problem. Solving the problem using the given information should lead to a height greater than 18.4 ft.

Well, Bobby sure knows how to have a "degree" of fun with angles, doesn't he? Let's get to the lighthouse math!

We have two angles, 30 degrees and 25 degrees, and a distance of 40 feet. To solve for the height of the lighthouse, we can use a bit of trigonometry, specifically the tangent function.

We'll assume that Bobby is standing on flat ground, not on a hill or in quicksand. If that's the case, we can create a right triangle with the height of the lighthouse as the unknown side, the distance Bobby backed up (40 feet) as the adjacent side, and the difference in the angles (30 - 25 = 5 degrees) as the opposite side.

Using the tangent function, we can set up the equation:

tan(5 degrees) = height of the lighthouse / 40 feet

Now let's do some calculations, shall we? Calculators at the ready!

tan(5 degrees) = 0.087
0.087 = height of the lighthouse / 40 feet

To solve for the height of the lighthouse, we can multiply both sides of the equation by 40:

0.087 * 40 feet = height of the lighthouse

So, with my mathematical meanderings complete, it seems that the height of the lighthouse is approximately 3.48 feet.

Of course, this is all assuming Bobby's measurements and trigonometric prowess are on point, but hey, that's the fun in math and humor - a delightful blend of accuracy and playfulness!

To solve the height of the lighthouse, we can use the concept of trigonometry, specifically the tangent function.

Let's denote the height of the lighthouse as 'h' and the distance between the initial position of Bobby and the lighthouse as 'x'.

In the first measurement, when Bobby is closer to the lighthouse, we can form a right triangle with the height of the lighthouse as the opposite side, 'h', and the distance 'x' as the adjacent side. The angle formed is 30 degrees.

Using the tangent function, we can write:

tan(30 degrees) = h / x

Now, in the second measurement, Bobby backs up 40 feet, so the distance between Bobby's second position and the lighthouse is 'x + 40'.

Again forming a right triangle with the height of the lighthouse as the opposite side, 'h', and the distance 'x + 40' as the adjacent side. The angle formed is 25 degrees.

Using the tangent function again, we can write:

tan(25 degrees) = h / (x + 40)

We now have a system of two equations:

tan(30 degrees) = h / x

tan(25 degrees) = h / (x + 40)

To solve for 'h', we can solve these equations simultaneously.

We can rearrange the first equation to solve for 'h':

h = x * tan(30 degrees)

Now we substitute this expression for 'h' in the second equation:

tan(25 degrees) = (x * tan(30 degrees)) / (x + 40)

Now we can solve this equation to find the value of 'x', and then substitute this value back into the expression for 'h' to determine the height of the lighthouse.

Using a calculator or trigonometric tables, we can evaluate the tangent of 30 degrees and 25 degrees to find the respective values. By substituting these values into the equation, we can solve for 'x'.

Once 'x' is determined, we substitute it back into the expression for 'h' to find the height of the lighthouse.

Label the original position as A and the second position as B, top of the lighhouse as P and its base as Q. ABQ is a straight line, AB = 40, angle B=25°, angle PAQ = 30°, AQ = x, and PQ = h

I used to teach this question is 2 ways:

1.
In triangle PBA, angle PAB = 150°
then angle BPA = 5°
by the sine law:
PA/sin25 = 40/sin5°
PA = 40sin25/sin5

in the right-angled triangle PAQ,
sin30° = h/PA
h = PAsin30 = (40sin25/sin5)(sin30) = 96.98..

method2:
in triangle PAQ, tan30 = h/x
h = xtan30
in triangle PBQ, tan25 = h/(x+40)
h = (x+40)tan25

thus: xtan30 = (x+40)tan25
xtan30 = xtan25 + 40tan25
x(tan30 - tan25) = 40tan25
x = 40tan25/(tan30-tan25)

h = xtan30
= (40tan25)(tan30)/(tan30-tan25)
= 96.98..

I consider method 1 as the easier of the two