Question

To measure the height of the cloud cover at an airport, a worker shines a spotlight upward at an angle 75° from the horizontal. an observer at a distance d = 560 m away measures the angle of elevation to the spot of light to be 45°. find the height h of the cloud cover,

Answer 1

Answer:

The height of the cloud cover is 441.66 meters

Step-by-step explanation:

Distance = 560 m

The height of the cloud cover = h meters

According to the diagram, the worker stands at point R,

Let RT = x

tan 45⁰ = $\frac{opposite}{adjacent}$ = $\frac{h}{x}$

therefore, 1 =  $\frac{h}{x}$, h = x

Then tan 75⁰ = $\frac{h}{560-x}$, substituting x = h, we have

3.732 = $\frac{h}{560-h}$

3.732(560 - h) =  h

3.732 × 560 = 3.732h + h

2089.92 = 4.732h

h = 441.66 m

Answer 2

The height of the cloud cover is 228.62 meters

The given parameters are:

$\mathbf{\alpha = 75^o}$

$\mathbf{\theta = 45^o}$

$\mathbf{d = 560m}$

See attachment for the image of the cloud cover.

From the attached image, we have the following sine ratios:

$\mathbf{sin(75) = \frac hx}$

$\mathbf{sin(45) = \frac h{560 - x}}$

Make h the subject in both equations

$\mathbf{ h = xsin(75)}$

$\mathbf{ h = (560 - x) sin(45)}$

So, we have:

$\mathbf{ xsin(75) = (560 - x) sin(45)}$

Open brackets

$\mathbf{ xsin(75) = 560sin(45) - x sin(45)}$

Collect like terms

$\mathbf{ xsin(75) + x sin(45)= 560sin(45) }$

Evaluate sine 45 and 75

$\mathbf{ 0.9659x + 0.7071x= 560 \times 0.7071}$

$\mathbf{ 1.673x= 395.976}$

Divide both sides by 1.673

$\mathbf{ x= 236.69}$

Recall that:

$\mathbf{ h = xsin(75)}$

So, we have:

$\mathbf{h = 236.69 \times 0.9659}$

$\mathbf{h = 228.618871}$

Approximate

$\mathbf{h = 228.62}$

Hence, the height of the cloud cover is 228.62 meters

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