Find area of the triangle with vertices at the point given in each of the following:
I. (1,0),(6,0),(4,3)
II. (2,7),(1,1),(10,8)
III. (−2,−3),(3,2),(−1,−8)
(i) The area of the triangle with vertices (1, 0), (6, 0), (4, 3) is given by the relation,
\(\triangle\)=\(\frac{1}{2}\)\(\begin{vmatrix}1&0&1\\6&0&1\\4&3&1\end{vmatrix}\)
=\(\frac{1}{2}\)[1(0-3)-0(6-4)+1(18-0)]
=\(\frac{1}{2}\)[-3+18]=\(\frac{15}{2}\) square units
(ii) The area of the triangle with vertices (2, 7), (1, 1), (10, 8) is given by the relation,
△=\(\frac{1}{2}\)\(\begin{vmatrix}2&7&1\\1&1&1\\10&8&1\end{vmatrix}\)
=\(\frac{1}{2}\) [2(1-8)-7(1-10)+1(8-10)]
=\(\frac{1}{2}\)[-14+63-2]
=\(\frac{1}{2}\)[-16+63]
=\(\frac{47}{2}\) square units
(iii) The area of the triangle with vertices (−2, −3), (3, 2), (−1, −8)
is given by the relation,
△=\(\frac{1}{2}\)\(\begin{vmatrix}-2&-3&1\\3&2&1\\-1&-8&1\end{vmatrix}\)
=\(\frac{1}{2}\)[-2(2+8)+3(3+1)+1(-24+2)]
=\(\frac{1}{2}\)[-20+12-22]
=-\(\frac{30}{2}\)=-15
Hence, the area of the triangle is I-15I=15 square units.
A settling chamber is used for the removal of discrete particulate matter from air with the following conditions. Horizontal velocity of air = 0.2 m/s; Temperature of air stream = 77°C; Specific gravity of particle to be removed = 2.65; Chamber length = 12 m; Chamber height = 2 m; Viscosity of air at 77°C = 2.1 × 10\(^{-5}\) kg/m·s; Acceleration due to gravity (g) = 9.81 m/s²; Density of air at 77°C = 1.0 kg/m³; Assume the density of water as 1000 kg/m³ and Laminar condition exists in the chamber.
The minimum size of particle that will be removed with 100% efficiency in the settling chamber (in $\mu$m is .......... (round off to one decimal place).