Question

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 yup at a 60 degree angle to see the same hot air ballon. approximate how far is the hot air ballon off the ground?

Answer 1

check the picture below.

thus then

$\bf \begin{cases} y=100-a\\ y=a\sqrt{3} \end{cases}\implies 100-a=a\sqrt{3}\implies 100=a+a\sqrt{3} \\\\\\ 100=a(1+\sqrt{3})\implies \boxed{\cfrac{100}{1+\sqrt{3}}=a} \\\\\\ \textit{what is the altitude \underline{y} then?}\qquad y=100-a \\\\\\ y=100-\cfrac{100}{1+\sqrt{3}}\implies y\approx 63.39745962156~yards$

Answer 2

The height of the air balloon is 63.4 yards.

What is a trignometry?

Trigonometry is a branch of mathematics that uses variables to determine heights and distances. It is the study of the properties of right angled triangles and trigonometric functions and of their applications.

For the given situation,

The diagram below shows the situation described above.

The 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 height of the air balloon be 'h'

⇒ $x+y=100$

⇒ $x=100-y$

We know that, $tan$ θ = $\frac{perpendicular }{base}$

Now consider the triangle with angle 45°,

$tan 45$° = $\frac{h}{x}$

We know that tan 45° = 1 and substitute x = 100-y,

⇒ $1 = \frac{h}{100-y}$

⇒ $h=100-y$

This is equation 1.

Now consider the triangle with angle 60°,

$tan60$° = $\frac{h}{y}$

We know that, tan 60° = √3

⇒ $\sqrt{3} = \frac{h}{y}$

⇒ $h=y\sqrt{3}$

This is the equation 2.

On equating equation 1 and 2,

⇒ $100-y=y\sqrt{3}$

⇒ $y+y\sqrt{3}=100$

⇒ $y(\sqrt{3}+1 )=100$

⇒ $y=\frac{100}{\sqrt{3}+1 }$

Thus height, $h=y\sqrt{3}$

Substitute the value of y in h,

⇒ $h=\sqrt{3}(\frac{100}{\sqrt{3} +1} )$

⇒ $h=\frac{100\sqrt{3} }{\sqrt{3}+1 }$

⇒ $h=\frac{173.205}{2.732}$

⇒ $h=63.397$

⇒ $h=63.4$

Hence we can conclude that the height of the air balloon is 63.4 yards.

Learn more about trigonometric ratios here

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