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Question
The height of a tower is 100 m. When the angle of elevation of the sun changes from 30° to 45°, the shadow of the tower becomes x metres less. The value of x is
Options
100 m
\[100\sqrt{3} m\]
\[100\left( \sqrt{3} - 1 \right) m\]
\[\frac{100}{3}m\]
Solution
The given situation can be represented as,
Here, AB is the tower of height 100 meters.
When angle of elevation of sun changes from`∠D=30°` to `∠C=45°`, .`CD=x`
We assumed that `BC=y`
Here we have to find the value of x
So we use trigonometric ratios.
In a triangle,`ABC`
`⇒ tan C=(AB)/(BC)`
`⇒ tan 45°=(AB)/(BC)`
`⇒1=100/y`
`⇒y=100`
Again in a triangle ABD,
`⇒ tan D=( AB)/(BC+CD)`
`⇒ tan 30°=100/(x+y)`
`⇒ 1/sqrt3=100/(x+y)`
`⇒ 100sqrt3=x+y`
`⇒100sqrt3=x+100` `Put x=100`
`⇒x=100(sqrt3-1)`
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