The length of the shadow of a pole on the level ground increases by 60 meter when one angle of elevation of the sun changes 54° to 32°.calaulate the height of the pole,correct to three significants figures.

over in the triangle between shadows we have a 32 degree angle. At the other end of our 60 meters we have a 180=54 = 126 degree angle. Therfore up at the top of that triangle we have a 180 - 32 - 126 = 22 degree angle

law of sines for long hypotenuse ray call it x
sin 126/x = sin 22/60
x = 60 sin 126/sin 22 = 130 meters
so
sin 32 = h /130
h = 68.7 meters high

Good

But no diagram

To solve this problem, we'll use the trigonometric relationship between the angle of elevation, the length of the shadow, and the height of the pole.

Let's denote the length of the shadow of the pole as "x" and the height of the pole as "h".

We are given that the length of the shadow increases by 60 meters when the angle of elevation changes.

From the trigonometric relationship, we know that:

tan(angle) = opposite side / adjacent side

Using this relationship, we can write two equations:

Equation 1: tan(54°) = h / x (for the initial angle of elevation)
Equation 2: tan(32°) = h / (x + 60) (for the changed angle of elevation)

Now, let's solve these equations to find the values of "h" and "x".

Equation 1: h = x * tan(54°) (equation obtained by rearranging Equation 1)
Equation 2: h = (x + 60) * tan(32°) (equation obtained by rearranging Equation 2)

Setting the right sides of both equations equal to each other, we can solve for "x":

x * tan(54°) = (x + 60) * tan(32°)
x * tan(54°) = x * tan(32°) + 60 * tan(32°)
x * [tan(54°) - tan(32°)] = 60 * tan(32°)
x = (60 * tan(32°)) / [tan(54°) - tan(32°)]

Using a scientific calculator or trigonometric table, we can find the values of tan(32°) ≈ 0.6249 and tan(54°) ≈ 1.3764.

Substituting these values into the equation, we get:

x = (60 * 0.6249) / (1.3764 - 0.6249)
x ≈ 34.4064

The length of the shadow, x, is approximately 34.4064 meters.

Now, let's substitute this value back into one of the equations to find the height of the pole:

h = x * tan(54°)
h = 34.4064 * tan(54°)
h ≈ 42.6974

The height of the pole, h, is approximately 42.6974 meters.

Therefore, the height of the pole is approximately 42.7 meters (rounded to three significant figures).

To find the height of the pole, we can use trigonometry. Let's denote the height of the pole as 'h'.

We are given two angles of elevation and the corresponding change in the length of the shadow. From this information, we can form a right triangle. The vertical side of the triangle represents the height of the pole (h), and the horizontal side represents the length of the shadow.

Let's label the triangle with the given information:
- The angle of elevation of 54° is opposite the side of length 'S'.
- The angle of elevation of 32° is opposite the side of length 'S + 60' (increase in shadow length).

Now, using trigonometric ratios, let's write the equations for each angle:
For the angle of 54°:
tan 54° = h / S

For the angle of 32°:
tan 32° = h / (S + 60)

Now we have two equations and two variables (h and S), so we can solve the system of equations to find the height of the pole.

Rearrange the first equation to solve for S:
S = h / tan 54°

Substitute this value of S into the second equation:
tan 32° = h / (h / tan 54° + 60)

Now we can solve this equation to find the value of h.

Using trigonometric identities:
tan 32° = h / (h / (tan 54°) + 60)
tan 32° = h * tan 54° / (h + 60 * tan 54°)

Multiply both sides by (h + 60 * tan 54°):
h * tan 32° + 60 * tan 32° * tan 54° = h * tan 54°

Rearrange the equation to solve for h:
h * (tan 54° - tan 32°) = 60 * tan 32° * tan 54°
h = (60 * tan 32° * tan 54°) / (tan 54° - tan 32°)

Now, plug in the values and calculate:
h = (60 * tan 32° * tan 54°) / (tan 54° - tan 32°)
h ≈ 138.773 (approximately)

Therefore, the height of the pole is approximately 138.773 meters, correct to three significant figures.