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Mathematics Assignment

An open container is such that each horizontal cross-section is an equilateral triangle. Its base has sides of length 20 cm, and its top has sides of length 10x cm. Each sloping edge has length 10 cm. The surface of the container is modelled by part of an inverted triangular pyramid, as shown below.
A rocket is modeled by a particle which moves along a vertical line. From launch, the rocket rises until its motor cuts after 18 seconds. At this time, it has reached a height of 540 metres above the launch pad and attained an upward velocity of 60 m/s. From this time on, the rocket has constant acceleration of -10m/s2 (due to the effect of gravity alone).
Let be the initial velocity of the rocket. Note that. Since the rocket is pointing directly upwards, we use the radian measure. Let be the position vector, given by where i and j are unit vectors. Also, note that .
Thus, and . (Note that 540 m is added since we have started from that point where the rocket motor runs out.)

Substituting for y, we obtain

Thus, the maximum height the rocket can reach is 720 metres.

(b) How long (from launch) does the rocket take to reach this maximum height
From (a), the rocket will reach its maximum height after 6 seconds.
(c) After how long (from launch) does the rocket crash onto the launch pad
When the rocket reaches its maximum height at 720 m, the velocity will then return to zero. Thus, .
Hence,

Thus, the rocket will crash onto the la ...Show more

Summary

(i) According to the First Derivative Test, if, for a differentiable function f, is increasing for all x in a neighbourhood of from the left and is decreasing for all x in a neighbourhood of from the right, then f has a relative maximum at. If is decreasing for all x in a neighbourhood of from the left and is increasing for all x in a neighbourhood of from the right, then f has a relative minimum at…
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