A billboard at the top of a building is being illuminated by a
projection light that is 102 feet from the base of the
building as shown in the figure. Determine how tall the
billboard if its lower and the upper ends makes angles of
and with the horizontal line from the projection light. Also
find the area of the triangle illuminated by the project light.

Ask by Tucker Pollard. in Puerto Rico

Nov 02,2024

Question

A billboard at the top of a building is being illuminated by a
projection light that is 102 feet from the base of the
building as shown in the figure. Determine how tall the
billboard if its lower and the upper ends makes angles of
and with the horizontal line from the projection light. Also
find the area of the triangle illuminated by the project light.

Ask by Tucker Pollard. in Puerto Rico

Nov 02,2024

UpStudy AI · Accepted Answer

To determine the height of the billboard and the area of the triangle illuminated by the projection light, we can use trigonometric relationships.

### Step 1: Determine the height of the billboard

Let:
- $$ h_1 $$ be the height of the lower end of the billboard,
- $$ h_2 $$ be the height of the upper end of the billboard,
- $$ d = 102 $$ feet (the distance from the projection light to the base of the building),
- $$ \theta_1 = 36^\circ $$ (angle for the lower end),
- $$ \theta_2 = 46^\circ $$ (angle for the upper end).

Using the tangent function, we can find the heights:

$$
h_1 = d \cdot \tan(\theta_1) = 102 \cdot \tan(36^\circ)
$$
$$
h_2 = d \cdot \tan(\theta_2) = 102 \cdot \tan(46^\circ)
$$

Now, we calculate $$ h_1 $$ and $$ h_2 $$:

1. Calculate $$ \tan(36^\circ) $$:
   $$
   \tan(36^\circ) \approx 0.7265
   $$
   Thus,
   $$
   h_1 = 102 \cdot 0.7265 \approx 74.0 \text{ feet}
   $$

2. Calculate $$ \tan(46^\circ) $$:
   $$
   \tan(46^\circ) \approx 1.0355
   $$
   Thus,
   $$
   h_2 = 102 \cdot 1.0355 \approx 105.6 \text{ feet}
   $$

### Step 2: Calculate the height of the billboard

The height of the billboard $$ H $$ is the difference between $$ h_2 $$ and $$ h_1 $$:

$$
H = h_2 - h_1 = 105.6 - 74.0 = 31.6 \text{ feet}
$$

### Step 3: Calculate the area of the triangle illuminated by the projection light

The area $$ A $$ of the triangle formed by the projection light and the two points on the billboard can be calculated using the formula for the area of a triangle:

$$
A = \frac{1}{2} \times \text{base} \times \text{height}
$$

Here, the base is the distance from the projection light to the base of the building, which is $$ d = 102 $$ feet, and the height is the difference in heights $$ H = 31.6 $$ feet.

Thus, the area is:

$$
A = \frac{1}{2} \times 102 \times 31.6
$$
$$
A = 51 \times 31.6 \approx 1615.6 \text{ square feet}
$$

### Summary of Results

- The height of the billboard is approximately $$ 31.6 $$ feet.
- The area of the triangle illuminated by the projection light is approximately $$ 1615.6 $$ square feet.
The height of the billboard is approximately 31.6 feet, and the area of the triangle illuminated by the projection light is approximately 1615.6 square feet.

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