Question

A man 1.5 height standing ontop of a vertical building 42m high,sees a truck some distance away,at an angle of depression of 53.5at what distance is the truck from the base of the building?

Answer 1

To find the distance of the truck from the base of the building, add the man's height to the building's height and divide it by the tangent of the angle of depression. This trigonometric approach will give the horizontal distance to the truck.

Step-by-step explanation:

The question involves using trigonometry to find the distance from a man standing on top of a building to a truck at ground level, given an angle of depression. The man's height of 1.5 meters is added to the building's height of 42 meters, giving a total height from which the angle of depression to the truck is measured. The angle of depression is congruent to the angle of elevation from the truck to the point directly below the man's eyes. To find the horizontal distance to the truck, the tangent function can be used:
Tangent(angle) = Opposite/Adjacent
Tangent(53.5°) = (42 + 1.5) meters / Distance
To solve for the Distance, we can rearrange the formula to:
Distance = (42 + 1.5) meters / Tangent(53.5°)
By taking the tangent of 53.5 degrees and dividing the combined height of the man and the building by this tangent, we can find the horizontal distance to the truck from the base of the building.

Answer 2

To solve this problem, we can use trigonometry. The tangent of an angle in a right triangle is the ratio of the opposite side to the adjacent side.

Let:

- \( h \) be the height of the man,

- \( H \) be the height of the building,

- \( d \) be the distance from the base of the building to the truck,

- \( \theta \) be the angle of depression.

In this case:

- \( h = 1.5 \) meters (height of the man),

- \( H = 42 \) meters (height of the building),

- \( \theta = 53.5^\circ \) (angle of depression).

The tangent of the angle of depression (\( \theta \)) is given by:

\[ \tan(\theta) = \frac{h}{d + H} \]

We can rearrange this equation to solve for \( d \):

\[ d = \frac{h}{\tan(\theta)} - H \]

Substitute the given values:

\[ d = \frac{1.5}{\tan(53.5^\circ)} - 42 \]

Now, calculate this expression to find the distance (\( d \)). Using a calculator:

\[ d \approx \frac{1.5}{\tan(53.5^\circ)} - 42 \approx 20.26 \]

So, the distance from the base of the building to the truck is approximately 20.26 meters.