Maths-
General
Easy

Question

# During mineral formation, the same chemical compound can be come different minerals depending on the temperature and pressure at the time of formation. A phase diagram is a graph that shows the conditions that are needed to form each mineral. The graph above is a portion of the phase diagram for aluminosilicates, with the temperature t , in degreesCelsius  , on the horizontal axis, and the pressure p , in gigapascals (G Pa), on theWhich of the following systems of inequalities best describes the region where sillimanite can form?

Hint:

### The boundaries of the region where sillimanite can be formed are given by two lines: one is the boundary line between kyanite and sillimanite and the other is the boundary between andalusite and sillimanite.First we find the equation of the boundary line between kyanite and sillimanite:There are two points lying on this line :  and .Let these points be denoted by andIn an xy plane, equation of a line passing through two points  (a , b ) and  (c , d ) is given byUsing the above points, we haveSimplifying the equation, we haveCross multiplying, we getWe use the calculator to expand the above equationTaking all the terms in the left hand side, we getDividing by 295 throught, we getAs the region which forms sillimanite is below this line, we change the equality to less thanor equal to inequality,Next,We find the equation of the boundary line between andalusite and sillimanite:There are two points lying on this line andLet these points be denoted byIn an xy plane, equation of a line passing through two points (a , b )  and (c, d )  is given byUsing the above points, we haveSimplifying the equation, we haveCross multiplying, we getWe use the calculator to expand the above equationTaking all the terms in the left hand side, we getDividing by 260 throught, we getAs the region which forms sillimanite is above this line, we change the equality to greater than or equal to inequality,Finally, we getThe inequalities which describe the region where sillimanite is formed areRewriting these inequalities, we getOption B)best describes the region where silliminate can form.

Note:
We do not have the exact inequalities in the option, so we choose the one from the options which is the closest approximation of the inequalities that we have calculated. There are a couple of formulas and concepts used here, such as the equation of a line passing through two points and the concept that the region below a line is given by replacing the equality sign with less than or equal to in the standard form of the equation and vice versa.