Alexa and Emma are looking up at their house from the backyard. From alexa's point of view, the top of the house is at an ang;e of elevation of 40 degree. From Emma's point of view, directly closer to the house, it is 60 degree. The house is 15 metres high. How far apart are the two girls?
Ah, the girls seem to be engaging in a little game of "Angle Wangle House Observation." Well, fear not, for I, Clown Bot, have arrived to bring some comedic confusion into the equation!
From Alexa's vantage point, we have an angle of elevation of 40 degrees, while Emma, being closer, has a greater perspective with a 60-degree angle of elevation. Quite the conundrum, isn't it?
To solve this puzzle, let's first calculate the distance between Alexa and the house. We can use a bit of trigonometric tomfoolery!
Since the height of the house is 15 meters and Alexa's angle of elevation is 40 degrees, we can use the tangent function to calculate the distance between Alexa and the house.
tan(40 degrees) = (15 meters) / x
Solving for x (the distance between Alexa and the house), we get:
x = (15 meters) / tan(40 degrees)
Using our trusty calculator for this circus act, we find that x is approximately 14.91 meters.
Now, let's turn our attention to Emma. Her angle of elevation is 60 degrees, and she is directly closer to the house. To find the distance between Emma and the house, we can once again employ the trigonometric tricks!
tan(60 degrees) = (15 meters) / y
Here, y represents the distance between Emma and the house. Calculating y, we get:
y = (15 meters) / tan(60 degrees)
With a dazzling display of calculation acrobatics, we find that y is approximately 8.66 meters.
So, the distance between the two girls is the difference between their respective distances from the house:
Distance = |x - y| = |14.91 meters - 8.66 meters|
Performing this arithmetic performance, we find that the girls are approximately 6.25 meters apart.
And there you have it! The two girls are 6.25 meters apart, engaging in a delightful escapade of angle observations. I hope this answer brings a smile to your face, my dear interlocutor!
To solve this problem, we can use trigonometry and create a right triangle with the height of the house as the vertical leg and the horizontal distance between the two girls as the base. Let's label the distance between the two girls as 'x'.
From Alexa's perspective, the angle of elevation to the top of the house is 40 degrees. Hence, we have the following trigonometric relationship:
tan(40°) = height of the house / distance between Alexa and the house
tan(40°) = 15 / x
From Emma's perspective, the angle of elevation to the top of the house is 60 degrees. Hence, we have the following trigonometric relationship:
tan(60°) = height of the house / (distance between Emma and the house - x)
tan(60°) = 15 / (x - distance between the two girls)
Now, we can solve the two equations simultaneously to find the value of 'x'.
First, solve the equation from Alexa's perspective:
tan(40°) = 15 / x
Rearrange the equation to solve for 'x':
x = 15 / tan(40°)
Calculate the value of 'x':
x ≈ 15 / 0.8391
x ≈ 17.870 meters
So, the distance between the two girls, Alexa and Emma, is approximately 17.870 meters.
To find the distance between the two girls, we can use the concept of trigonometry and the angle of elevation.
Let's label the distance between Alexa and the house as 'a' and the distance between Emma and the house as 'b'. We need to find the value of 'a + b', which gives us the distance between the two girls.
To begin, we can use the trigonometric relationship between the angle of elevation and the height of the house. In this case, we can consider the right triangle formed by the viewer (Alexa or Emma), the top of the house, and the base of the house.
Using the angle of elevation of 40 degrees for Alexa and 60 degrees for Emma, we can apply the tangent function, which is defined as the ratio of the opposite side to the adjacent side of a right triangle.
For Alexa:
tan(40 degrees) = Height of the house (15 meters) / distance 'a'
For Emma:
tan(60 degrees) = Height of the house (15 meters) / distance 'b'
Now, we need to solve for 'a' and 'b' separately.
For Alexa:
a = Height of the house (15 meters) / tan(40 degrees)
For Emma:
b = Height of the house (15 meters) / tan(60 degrees)
Calculating each value:
a ≈ 15 / tan(40) ≈ 15 / 0.8391 ≈ 17.87 meters
b ≈ 15 / tan(60) ≈ 15 / 1.732 ≈ 8.66 meters
Finally, to find the distance between Alexa and Emma, we add 'a' and 'b':
Distance = a + b
Distance ≈ 17.87 + 8.66 ≈ 26.53 meters
Therefore, the two girls are approximately 26.53 meters apart.
We draw 2 rt. triangles with a common
ver. side and a shared hor. side with
separate hyp:
X = hor. = X1 + X2 = Alexa's dist. from
the house.
X1 = Emma's dist. from the house.
X2 = The dist. between the girls.
Y = Ver. = 15 m.
tan60 = Y/X1 = 15/X1.
X1 = 15/tan60 = 8.66 m.
tan40 = Y/(X1+X2).
tan40 = 15/(8.66+X2)
8.66+X2 = 15/tan40 = 17.88
X2 = 17.88 - 8.66 = 9.22 m. = Dist.
between the girls.