[Solved] Caleb and Emily are standing 100 yards from each other. Caleb looks up at a 45-degree angle to see a hot air balloon. Emily looks up at a 60-degree angle to see the same hot air balloon. Approximate how far is the hot air balloon off the ground.

BySolutionlogy

Verified Answer: Caleb and Emily are standing 100 yards from each other. Caleb looks up at a 45-degree angle to see a hot air balloon. Emily looks up at a 60-degree angle to see the same hot air balloon. Approximate how far is the hot air balloon off the ground.

Solution: According to the question, it is given that;

Distance between Caleb and Emily = 100 yards

Let the distance from Caleb to the air balloon be = x

Let the distance from Emily to the air balloon be = y

Let the height of the air balloon be = h

As per the question,

Distance from Caleb to the air balloon + Distance from Emily to the air balloon=100 yards

x + y = 100

⟹x = 100 – y

Welcome To MathchaWe know that tan𝜃=Perpendicular
Base Let us consider the triangle in figure 1 with angle 45°, we get tan45°=h
x ——
1
 We know that tan45°=1 and x=100–y. Substuting the value of tan45° and x in equation
1
, we get 1=h
100–y h=100–y——
2
 Now, Let us consider the triangle in figure 1 with angle 60°, we get tan60°=h
y ——
3
 We know that tan60°=3. Substuting the value of tan60° in equation
1
, we get 3=h
y h=y3——
4
 Comparing equation
2
 and
4
, we get 100–y=y3 y+y3=100 y=100
3+1 Now, substuting the value of y in equation
4
, we get h=3(100
3+1) h=1003
3+1 h=173.205
2.732 h=63.397≃63.4 ⟹h=63.4

Hence, the height of the air balloon is 63.4 yards.

Conclusion

Caleb and Emily are standing 100 yards from each other. Caleb looks up at a 45° angle to see a hot air balloon. Emily looks up at a 60° angle to see the same hot air balloon. The hot air balloon is 63.4 yards off the ground.

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