(A) Let the height of the tower be $h = 5 \text{ m}$. Let the base of the tower be $O$. Let the position of the first person be $A$ (South) and the se

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(A) Let the height of the tower be $h = 5 \text{ m}$. Let the base of the tower be $O$. Let the position of the first person be $A$ (South) and the second person be $B$ (West).In $\triangle POA$,$\tan(45^{\circ}) = \frac{PO}{OA} \implies 1 = \frac{5}{OA} \implies OA = 5 \text{ m}$.In $\triangle POB$,$\tan(30^{\circ}) = \frac{PO}{OB} \implies \frac{1}{\sqrt{3}} = \frac{5}{OB} \implies OB = 5\sqrt{3} \text{ m}$.Since the directions South and West are perpendicular,$\triangle AOB$ is a right-angled triangle at $O$.The distance between the two persons is $AB = \sqrt{OA^2 + OB^2} = \sqrt{5^2 + (5\sqrt{3})^2} = \sqrt{25 + 75} = \sqrt{100} = 10 \text{ m}$.

Class 11 Mathematics Trigonometrical Equations Height and Distance

Medium

The angle of elevation of the top $P$ of a tower from the feet of one person standing due South of the tower is $45^{\circ}$ and from the feet of another person standing due West of the tower is $30^{\circ}$. If the height of the tower is $5 \text{ m}$,then the distance (in meters) between the two persons is equal to $..........$.

JEE MAIN 2023 All JEE MAIN

A
$10$
B
$5$
C
$5 \sqrt{5}$
D
$5 \sqrt{2}$

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