Respuesta :
Answer:
The distance between Tom and Sam is about 352.37m.
Step-by-step explanation:
Previous explanation
We can answer this question using Trigonometry.
Firstly, we need to draw all the information given in the question:
- Tom and Sam are on the opposite sides of a building and outside it.
- The building's height is 160 meters.
- The angles of elevation are measured from the positions Tom and Sam are. They are 40 and 55 degrees, respectively. We can see if we draw a line from Tom's position to the top of the building, we can form a triangle, as depicted in the graph below. The same applies to Sam's position (that is, we can form another triangle).
- The building is 50 meters wide, which completes the needed information to solve the question.
We attached a graph containing all this information (see below).
Solution
Having the elevation angles, the building height and width, from graph below, we can determine that the total distance is:
[tex] \\ d_{T} = d_{Tom} + 50m + d_{Sam}[/tex]
Where
[tex] \\ d_{Tom}\;is\;the\;Tom's\;distance\;from\;the\;building[/tex].
[tex] \\ d_{Sam}\;is\;the\;Sam's\;distance\;from\;the\;building[/tex].
Knowing from Trigonometry that the tangent of an angle is defined as:
[tex] \\ tan(\theta) = \frac{opposite\;side\;of\;\theta}{adjacent\;side\;of\;\theta}[/tex]
Then
[tex] \\ tan(\alpha) = \frac{opposite\;side\;of\;\alpha}{adjacent\;side\;of\;\alpha}[/tex]
And
[tex] \\ tan(\beta) = \frac{opposite\;side\;of\;\beta}{adjacent\;side\;of\;\beta}[/tex]
The opposite side of [tex] \\ \alpha[/tex] is the height of the building (160m). The same is the case for the opposite side of [tex] \\ \beta[/tex], that is, the height of the building too.
The adjacent side for [tex] \\ \alpha[/tex] is [tex] \\ d_{Tom}[/tex] and for [tex] \\ \beta[/tex] is [tex] \\ d_{Sam}[/tex].
The value of the tangent for an angle (alpha, in this case) = 40 degrees, rounding it to two decimals places is about: tan(40) = 0.84.
The value of the tangent for an angle (beta, in this case)= 55 degrees, rounding it to two decimals places is approximately: tan(55) = 1.43.
Then
[tex] \\ tan(\alpha) = \frac{160m}{d_{Tom}}[/tex]
Solving the equation for the distance from Tom to the building:
[tex] \\ tan(\alpha) = \frac{160m}{d_{Tom}}[/tex]
[tex] \\ d_{Tom} = \frac{160m}{tan(40)}[/tex]
[tex] \\ d_{Tom} = \frac{160m}{0.84}[/tex]
[tex] \\ d_{Tom} = \frac{160m}{0.84} \approx 190.48m[/tex]
For calculating Sam's distance from the building, we can proceed following the same steps:
[tex] \\ tan(\beta) = \frac{160m}{d_{Sam}}[/tex]
[tex] \\ d_{Sam} = \frac{160m}{tan(\beta)}[/tex]
[tex] \\ d_{Sam} = \frac{160m}{1.43} \approx 111.89m[/tex]
So
The distance between Tom and Sam is the resulting value of the sums:
[tex] \\ d_{T} = d_{Tom} + 50m + d_{Sam}[/tex]
[tex] \\ d_{T} = 190.48m + 50m + 111.89m[/tex]
[tex] \\ d_{T} = 352.37m[/tex]
Thus, The distance between Tom and Sam is about 352.37m.
See the graph below.
(Note: without rounding the respectives values for tangent of each angle, the value for the distance is about 352.71m).
