Calculus  Differentiation  Applied max/min questions.
Type 6: Rates  Miscellaneous questions.
1. At the entrance to a mine there are electricity wires.
The height (H) of the wires in metres above the ground at the mine entrance varies with the outside temperature (t) in degrees Celsius and is given by When the temperature outside the mine reaches 38° C, all work must stop. Answer.(i) H = 21.82 m. (ii) Max temperature is 34.5° C. 

2. A light is to be placed over the centre of a circle, radius a units.
The intensity I of the light varies as the sine of the angle α at which the rays strike the illuminated surface, divided by the square of the distance d from the light. This relationship can be expressed mathematically as . Answer.Max height when y = a / √2. 

3. At a music concert, speakers are placed in two towers (S1 and S2) which are 75 metres apart. The intensity of sound (I) produced by a speaker tower of power P at a distance x metres from the tower is given by
. The speakers in tower S_{1} have a power output of P but the older speakers in tower S_{2} produce 25% less power. The diagram shows Grace stands at G between the two towers and she is x metres from tower S_{1}. (i) Show that the total sound intensity from both speaker towers I_{T} at point G is (ii) Given that the total sound intensity from both speaker towers is I_{T} at point G, find the derivative . (iii) Hence find the value of x (to the nearest half metre) which minimises the sound intensity so that Grace knows where best to stand so she can enjoy the concert. Answer.Grace stands 39.3 m from S_{1}. 

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