The angle elevation from a point A to the top of the Washington Monument is 32 degrees. From point B, which is not the same line but 55 feet closer to the monument, the angle elevation to the top is 38 degrees. Find the height of the Washington Monument.

Is it 1.27

Angle in point A = 32 °

tan 32 ° = H / L

L = H / tan 32 ° = H * ctg 32 °

Angle in point B = 38 °

tan 38 ° = H / ( L - 55 )

L - 55 = H / tan 38 ° = H * ctg 38 °

L - 55 = H * ctg 38 ° Add 55 to both sides

L - 55 + 55 = H * ctg 38 + 55

L = H * ctg 38 ° + 55

L = L

H * ctg 32 ° = H * ctg 38 ° + 55 Subtract H * ctg 38 ° to both sides

H * ctg 32 ° - H * ctg 38 ° = H * ctg 38 ° + 55 - H * ctg 38 °

H * ctg 32 ° - H * ctg 38 ° = 55

H * ( ctg 32 ° - ctg 38 ° ) = 55 Divide both sides by ( ctg 32 ° - ctg 38 ° )

H = 55 / ( ctg 32 ° - ctg 38 ° )

H = 55 / ( 1.60033 - 1.27994 )

H = 55 / 0.32039

H = 171.666

But in w i k i p e d i a you can find that Washington Monument standing 555 feet 5 1 ⁄ 8 inches (169.294 m) tall.

Probably measuring unit is the meter, not the feet.

Remark:

ctg is cotangent ( cot )

I use a slightly different method.

In my sketch , I labeled the top of the monument P and the bottom Q
By properties of the sum of angles in a triangle,
in triangle ABP , angle A = 32°, angle ABP = 142, and angle APB = 6°

using the Sine Law:
BP/sin32 = 55/sin6
BP = 55sin32/sin6 = 278.8289..

now in the right-angle triangle PBQ
sin 38° = PQ/BP
PQ = 278.8289..(sin38) = 171.6642...

the monument is appr 171.7 ft heigh

Reiny,

Are you sure that the height of the Washington Monument is 171.7 FEET?

Bosnian, thanks for catching that

The question used feet as its units, and I just did the math without really thinking about that height.
I think the linear units should have been metres and not feet.
Both of our numerical values were correct, you also had 171.7

I recall taking the elevator up back in the 60's and then walking down the stairs of the monument. Very interesting to see the huge stones from the inside with their place of origin chiseled into them

To solve this problem, we can use the concept of trigonometry and set up a proportion based on the given information.

Let's consider a right triangle formed by the Washington Monument, point A, and a point C directly below the top of the monument. The angle of elevation from point A to the top of the monument is given as 32 degrees.

Similarly, let's consider another right triangle formed by the Washington Monument, point B, and the same point C directly below the top. The angle of elevation from point B to the top of the monument is given as 38 degrees.

We need to find the height of the Washington Monument, which is the length of segment AC.

Since point B is 55 feet closer to the monument than point A, we can label the length of segment BC as 55 feet.

Now, let's set up a proportion using the tangent function:

tan(32 degrees) = AC / BC
tan(38 degrees) = AC / (BC - 55)

We can rearrange the first equation to express AC in terms of BC:
AC = BC * tan(32 degrees)

Substituting this expression into the second equation, we can solve for BC:
tan(38 degrees) = (BC * tan(32 degrees)) / (BC - 55)

Now, we can solve for BC using algebra. Multiply both sides of the equation by (BC - 55):
(BC - 55) * tan(38 degrees) = BC * tan(32 degrees)

Expanding the equation:
BC * tan(38 degrees) - 55 * tan(38 degrees) = BC * tan(32 degrees)

Rearranging the equation to express BC terms on one side:
BC * (tan(38 degrees) - tan(32 degrees)) = 55 * tan(38 degrees)

Finally, we can solve for BC by dividing both sides of the equation by (tan(38 degrees) - tan(32 degrees)):
BC = (55 * tan(38 degrees)) / (tan(38 degrees) - tan(32 degrees))

Once we have the value of BC, we can substitute it back into the expression to find AC:
AC = BC * tan(32 degrees)

This will give us the height of the Washington Monument, which is the value of AC.