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A body is projected at such an angle that the horizontal range is three times the maximum height. The angle of projection is:
Range: $R = \dfrac{u^2\sin2\theta}{g}$, Max height: $H = \dfrac{u^2\sin^2\theta}{2g}$
Given $R = 3H$: $\dfrac{\sin2\theta}{1} = 3\cdot\dfrac{\sin^2\theta}{2} \Rightarrow 2\sin\theta\cos\theta = \dfrac{3}{2}\sin^2\theta$
$\Rightarrow \tan\theta = \dfrac{4}{3} \Rightarrow \theta = \tan^{-1}\!\left(\dfrac{4}{3}\right) \approx \mathbf{53°}$
If the arithmetic mean of 5 consecutive integers is 12, what is the sum of the least and greatest of these integers?
For 5 consecutive integers, the mean equals the middle (3rd) integer. So the middle integer is 12.
The 5 integers are: $10, 11, 12, 13, 14$.
The least is 10 and the greatest is 14.
Sum $= 10 + 14 = 24$.
Magnetic field inside a solenoid is:
Inside ideal solenoid, field is uniform and parallel to axis.
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