Tutor profile: Sai Sree Satya J.

A cargo plane has three compartments for storing cargo: front, centre and rear. These compartments have the following limits on both weight and space: Compartment Weight capacity (tonnes) Space capacity (cubic metres) Front 10 6800 Centre 16 8700 Rear 8 5300 Furthermore, the weight of the cargo in the respective compartments must be the same proportion of that compartment's weight capacity to maintain the balance of the plane. The following four cargoes are available for shipment on the next flight: Cargo Weight (tonnes) Volume (cubic metres/tonne) Profit (£/tonne) C1 18 480 310 C2 15 650 380 C3 23 580 350 C4 12 390 285 Any proportion of these cargoes can be accepted. The objective is to determine how much (if any) of each cargo C1, C2, C3 and C4 should be accepted and how to distribute each among the compartments so that the total profit for the flight is maximised. Formulate the above problem as a linear program

Variables: We need to decide how much of each of the four cargoes to put in each of the three compartments. Hence let: xij be the number of tonnes of cargo i (i=1,2,3,4 for C1, C2, C3 and C4 respectively) that is put into compartment j (j=1 for Front, j=2 for Centre and j=3 for Rear) where xij >=0 i=1,2,3,4; j=1,2,3 Note here that we are explicitly told we can split the cargoes into any proportions (fractions) that we like. Constraints: cannot pack more of each of the four cargoes than we have available x11 + x12 + x13 <= 18 x21 + x22 + x23 <= 15 x31 + x32 + x33 <= 23 x41 + x42 + x43 <= 12 the weight capacity of each compartment must be respected x11 + x21 + x31 + x41 <= 10 x12 + x22 + x32 + x42 <= 16 x13 + x23 + x33 + x43 <= 8 the volume (space) capacity of each compartment must be respected 480x11 + 650x21 + 580x31 + 390x41 <= 6800 480x12 + 650x22 + 580x32 + 390x42 <= 8700 480x13 + 650x23 + 580x33 + 390x43 <= 5300 the weight of the cargo in the respective compartments must be the same proportion of that compartment's weight capacity to maintain the balance of the plane [x11 + x21 + x31 + x41]/10 = [x12 + x22 + x32 + x42]/16 = [x13 + x23 + x33 + x43]/8 Objective The objective is to maximise total profit, i.e. maximise 310[x11+ x12+x13] + 380[x21+ x22+x23] + 350[x31+ x32+x33] + 285[x41+ x42+x43]

If $$\displaystyle \sin{{A}}+{{\sin}^{{{2}}}{A}}={1}$$ and $$\displaystyle{a}{{\cos}^{{{12}}}{A}}+{b}{{\cos}^{{{8}}}{A}}+{c}{{\cos}^{{{6}}}{A}}-{1}={0}$$ then $$\displaystyle{b}+\frac{c}{{a}}+{d}=?$$

$$\displaystyle \sin{{A}}={1}-{{\sin}^{{{2}}}{A}}$$ $$\displaystyle \sin{{A}}={{\cos}^{{{2}}}{A}}$$ $$\displaystyle{{\sin}^{{{2}}}{A}}={{\cos}^{{{4}}}{A}}$$ $$\displaystyle{1}-{{\cos}^{{{2}}}{A}}={{\cos}^{{{4}}}{A}}$$ $$\displaystyle{1}={{\cos}^{{{4}}}{A}}+{{\cos}^{{{2}}}{A}}$$ $$\displaystyle{1}^{{{3}}}={\left({{\cos}^{{{4}}}{A}}+{{\cos}^{{{2}}}{A}}\right)}^{{{3}}}$$ $$\displaystyle{1}={{\cos}^{{{12}}}{A}}+{3}{{\cos}^{{{10}}}{A}}+{3}{{\cos}^{{{8}}}{A}}+{{\cos}^{{{6}}}{A}}$$ Transverse 1 onto the other side we get the condition as given as in the question. Comparing the variables we get a=1, b=3, c=3, d=1 hence the value of $$\displaystyle{b}+\frac{c}{{a}}+{d}=$$ 3+3+1 = 7

integral of $$\frac{1}{1+\sin(x)}$$

$$\displaystyle{I}=\int\frac{1}{{{1}+ \sin{{x}}}}{\left.{d}{x}\right.}=\int\frac{{{\left({1}- \sin{{x}}\right)}}}{{{\left({1}+ \sin{{x}}\right)}{\left({1}- \sin{{x}}\right)}}}{\left.{d}{x}\right.}$$ $$\displaystyle\Rightarrow{I}=\int\frac{{{1}- \sin{{x}}}}{{{1}-{{\sin}^{2}{x}}}}{\left.{d}{x}\right.}$$ $$\displaystyle=\int\frac{{{1}- \sin{{x}}}}{{{\cos}^{2}{x}}}{\left.{d}{x}\right.}$$ $$\displaystyle=\int{\left[\frac{1}{{{\cos}^{2}{x}}}-\frac{ \sin{{x}}}{{{\cos}^{2}{x}}}\right]}{\left.{d}{x}\right.}$$ $$\displaystyle=\int{\left[{{\sec}^{2}{x}}- \sec{{x}} \tan{{x}}\right]}{\left.{d}{x}\right.}$$ $$\displaystyle{I}= \tan{{x}}- \sec{{x}}+{c}$$