(Solved): (a) An engincer is designing a right-angle triangular roof as shown in FIGURE Q3 (a) with one colu ...

(a) An engincer is designing a right-angle triangular roof as shown in FIGURE Q3 (a) with one column. In order to make sure that the roof is stable and safe, the centre of mass need to be calculated before the design stage. Determine mass, moment of mass and centre of mass of the roof. Assume that the density is \( x \). (8 marks) Bigure \( Q, 3 \) (a)e Right-angle triangulac noof (b) Estimate the vector value function \[ \mathbf{r}(t)=5 \cos (t) \mathbf{i}+5 \sin (t) \mathbf{j}+t \mathbf{k} \] if \( t=0 \) and \( t=\pi / 2 \). Then, sketch the graph of the vector value function. (5 marks) (c) Calculate the domain and limit \( (t \rightarrow 0) \) of the vector-valued function \( r(t)=\sqrt[4]{t} \mathbf{i}+ \) \( \sin 5 t \mathrm{j}+\ln 5 t \) (4 marks) (d) For the given function, \( \mathbf{r}(t)=\frac{1}{t} \mathbf{i}+\operatorname{tcos} 3 t \mathbf{j}+6 \mathbf{k} \) i) Calculate \( r^{\prime}(t) \) and \( \int r(t) \) (6 marks) ii) After integration, three constants are introduced for each vector component. Based on your opinion, are these three constants similar or different. Explain your answer. (2 marks)