The angle elevation from a point A to the top of the Washington Monument is 32'. From point B, which is on the same line but 55 feet closer to the monument, the angle of elevation to the top is 38'. Find the length of the Washington Monument.

I assume you mean degrees and not minutes

h is the height we want
x is distance to the monument from the closer point

tan 38 = h/x
tan 32 = h/(55+x)

x = h/tan 38 = 1.28 h

tan 32 = h/(55 + 1.28 h)

34.4 + .8 h = h

h = 172

To find the length of the Washington Monument, we need to use trigonometry.

Let's assume the height of the Washington Monument is "h" and the distance from point A to the monument is "x".

From point A, the angle of elevation to the top of the monument is 32'. This means that:

tan(32') = h / x

Similarly, from point B (which is 55 feet closer to the monument), the angle of elevation is 38'. This means that:

tan(38') = h / (x - 55)

Now we have a system of two equations with two variables. We can solve this system to find the values of "h" and "x". We will use the first equation to express "h" in terms of "x" and then substitute it into the second equation.

First, let's solve the first equation for "h":

h = x * tan(32')

Now, substitute the expression for "h" into the second equation:

tan(38') = (x * tan(32')) / (x - 55)

To solve for "x", multiply both sides of the equation by (x - 55) to get rid of the denominator:

(x - 55) * tan(38') = x * tan(32')

Expand the left side of the equation:

x * tan(38') - 55 * tan(38') = x * tan(32')

Rearrange the equation to have all the terms with "x" on one side:

x * tan(38') - x * tan(32') = 55 * tan(38')

Factor out "x":

x * (tan(38') - tan(32')) = 55 * tan(38')

Divide both sides by (tan(38') - tan(32')):

x = (55 * tan(38')) / (tan(38') - tan(32'))

Now that we have the value of "x", we can substitute it back into the equation for "h" to find its value:

h = x * tan(32')

Substitute the value of "x" we just found:

h = ((55 * tan(38')) / (tan(38') - tan(32'))) * tan(32')

Finally, we can calculate the height "h" of the Washington Monument using a scientific calculator.

To solve this problem, we can use the concept of similar triangles. Let's label the length from point A to the top of the monument as x.

We have two right triangles, one formed by point A and the top of the monument, and another formed by point B and the top of the monument.

From triangle AOB, where O is the top of the monument, we have the following information:
Angle AOB = 90 degrees
Angle AOM = 32 degrees, where M is the midpoint of AB
Length AM = x + 55 feet

From triangle BOA, we have the following information:
Angle BOA = 90 degrees
Angle BOM = 38 degrees
Length BM = x feet

Since the two triangles are similar, we can set up the following proportion:

(AM / BM) = (AO / BO)

Substituting the given lengths and angles, we have:

((x + 55) / x) = tan(38 degrees) / tan(32 degrees)

Now, we can solve for x to find the length of the Washington Monument.

First, calculate the tangent values for the angles:

tan(38 degrees) ≈ 0.78129
tan(32 degrees) ≈ 0.62487

We can now rewrite the proportion:

(x + 55) / x ≈ 0.78129 / 0.62487

Cross-multiplying, we get:

0.62487(x + 55) ≈ 0.78129(x)

Simplifying, we have:

0.62487x + 34.36835 ≈ 0.78129x

Rearranging the equation, we get:

0.78129x - 0.62487x ≈ 34.36835

0.15642x ≈ 34.36835

Dividing both sides by 0.15642, we find:

x ≈ 220.044

Therefore, the length of the Washington Monument is approximately 220.044 feet.