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The mosquito net over a \[7\;{\text{ft}} \times 4\;{\text{ft}}\]bed is $3\;{\text{ft}}$ high. The net has a hole at one corner of the bed through which a mosquito enters the net. Find the magnitude of the displacement of the mosquito.
A) $\sqrt {54} \;{\text{ft}}$
B) $81\;{\text{ft}}$
C) $\sqrt {74} \;{\text{ft}}$
D) $\sqrt {64} \;{\text{ft}}$

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Last updated date: 21st Apr 2024
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Answer
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Hint: The above problem is based on the kinematics. The displacement is defined as the change in the position and direction of the object. The net over the bed is the same as the box in the shape of a cube. The displacement of the mosquito can be found by the formula to find the length of the diagonal of the box.

Complete step by step answer
Given: The length of the bed is $l = 7\;{\text{ft}}$, width of the bed is $b = 4\;{\text{ft}}$ and height of the net is $h = 3\;{\text{ft}}$.
The formula to calculate the magnitude of the displacement of the mosquito is given as:
$d = \sqrt {{l^2} + {b^2} + {h^2}} $
Substitute $7\;{\text{ft}}$for l, $4\;{\text{ft}}$for b and $3\;{\text{ft}}$for h in the above formula to find the magnitude of the displacement of the mosquito.
$d = \sqrt {{{\left( {7\;{\text{ft}}} \right)}^2} + {{\left( {4\;{\text{ft}}} \right)}^2} + {{\left( {3\;{\text{ft}}} \right)}^2}} $
$d = \sqrt {74} \;{\text{ft}}$
Thus, the magnitude of the displacement of the mosquito is $\sqrt {74} \;{\text{ft}}$and the

option (A) is the correct answer.

Additional Information: The displacement and distance of the object is different. The displacement consists both magnitude and direction while distance only consists of magnitude. The change in the displacement of the object with time is the same as the velocity and change in the distance of the object with time is the same as the speed of the object.

Note: The displacement of the mosquito can also be found by using the vector method. The one corner of the net assumed as the origin and coordinates as (0, 0, 0) and the coordinates of the mosquito as (7 ft, 4 ft, 3 ft). Then find the distance between these two coordinates to find the magnitude of the mosquito.