50 POINTSSSS
From a point at the ground level infront of a tower, the angle of elevations of the top and bottom of flagstaff 6 m high situated at the top of a tower are observed 60° and 45° respectively. Find the height of the tower and the distance between the base of the tower and point of observation.

From the given information, we can form two right triangles. One triangle is formed by the observer, the top of the tower, and the top of the flagstaff. The other triangle is formed by the observer, the bottom of the tower, and the bottom of the flagstaff.

Let's solve for the height of the tower first:

In the triangle formed by the observer, the top of the tower, and the top of the flagstaff, we have:

tan(60°) = (h + 6) / d

Since tan(60°) = √3, we can rewrite the equation as:

√3 = (h + 6) / d

Multiplying both sides by d, we get:

√3d = h + 6

Next, in the triangle formed by the observer, the bottom of the tower, and the bottom of the flagstaff, we have:

tan(45°) = h / d

Since tan(45°) = 1, we can rewrite the equation as:

1 = h / d

Rearranging, we get:

h = d

Now, we can substitute this value of 'h' in the first equation:

√3d = d + 6

Simplifying, we have:

√3d - d = 6

Factoring out 'd', we get:

d (√3 - 1) = 6

Dividing both sides by (√3 - 1), we get:

d = 6 / (√3 - 1)

To find the height of the tower, we can substitute this value of 'd' in the equation h = d:

h = 6 / (√3 - 1)

Now we can calculate the values:

d ≈ 6 / (√3 - 1) ≈ 6 / (1.732 - 1) ≈ 6 / 0.732 ≈ 8.196 meters

h ≈ 6 / (√3 - 1) ≈ 6 / (1.732 - 1) ≈ 6 / 0.732 ≈ 8.196 meters

Therefore, the height of the tower is approximately 8.196 meters, and the distance between the base of the tower and the point of observation is also approximately 8.196 meters.

semsee45 semsee45 · Answer 2 · 2024-01-02T21:10:56+01:00

Answer:

Height of the tower = (3 + 3√3) m = 8.2 m (nearest tenth)
Distance between the base of the tower and the point of observation = (3 + 3√3) m = 8.2 m (nearest tenth)

Step-by-step explanation:

The given scenario can be modelled as two right triangles (see attached diagram). Given that a 6 m flagstaff is situated at the top of a tower, and the angles of elevations from a point of observation to the top and bottom of the flagstaff are 60° and 45° respectively, then:

The smaller right triangle has an angle of elevation of 45° and a height of y m.
The larger right triangle has an angle of elevation of 60° and a height of (y + 6) m.

Let x be the distance between the point of observation and the base of the tower. This is the base of both right triangles.

We can use the tangent trigonometric ratio to create two equations for the two right triangles:

[tex]\tan(45^{\circ})=\dfrac{y}{x}[/tex]

[tex]\tan(60^{\circ})=\dfrac{6+y}{x}[/tex]

As tan(45°) = 1 and tan(60°) = √3, then:

[tex]1=\dfrac{y}{x}[/tex]

[tex]\sqrt{3}=\dfrac{6+y}{x}[/tex]

Rearrange both equations to isolate x:

[tex]x=y[/tex]

[tex]x=\dfrac{6+y}{\sqrt{3}}[/tex]

Note that as x = y, the distance between the base of the tower and the point of observation and the height of the tower are equal.

Substitute the first equation into the second equation and solve for y:

[tex]\begin{aligned}y&=\dfrac{6+y}{\sqrt{3}}\\\\\sqrt{3}y&=6 + y\\\\\sqrt{3}y-y&=6\\\\y(\sqrt{3}-1)&=6\\\\y&=\dfrac{6}{\sqrt{3}-1}\\\\y&=3+3\sqrt{3}\end{aligned}[/tex]

Therefore the height (y) of the tower is (3 + 3√3) m, which is approximately 8.2 m (rounded to the nearest tenth).

As the height of the tower is equal to the distance between the base of the tower and the point of observation, then the distance is also (3 + 3√3) m, which is approximately 8.2 m (rounded to the nearest tenth).