The side of a hill is an inclined plane with slope of 1 in 30. A level railway running along the surface of the hill curves into it by a cutting which is 20 yards deep at the point at which the entrance to a tunnel is reached. The radius of the curve is 1000 yards. The cutting is 10 yards wide at the bottom, and the sides slope at an angle of 45\(^\circ\). Estimate the cubic contents of the earth excavated in making the cutting.
Prove that if \(a, b\), and \(c\) are positive integers the chance that \(a^2 + b^2 + c^2\) is divisible by 7 is one-seventh.
Two equal circular cylinders rest in parallel positions on a horizontal plane. An isosceles triangular prism, whose vertical angle is \(\alpha\), rests between them in a symmetrical position, its base being horizontal. If all the surfaces are equally rough, shew that equilibrium will be preserved if the coefficient of friction exceeds \(\tan \frac{1}{4}(\pi-\alpha)\).
Find the ratio of the volume of a regular tetrahedron to the volume of the regular tetrahedron formed by joining the central points of the faces of the former.
Prove the formula \[ f(x+h) - f(x) = hf'(x+\theta h), \] where \(0 < \theta < 1\). Deduce that a function of \(x\) whose differential coefficient is positive increases steadily with \(x\). Prove that the functions \[ \log(1+x) - \frac{x}{1+x}, \quad (2+x)\log(1+x) - 2x \] are positive for all positive values of \(x\); and shew how this sequence of functions may be continued further.
Prove that, if a series of polygons with a given number of sides are drawn with each side in a given direction, and all the angular points but one on specified straight lines, the locus of the last angular point is a straight line. Show how this may be a useful aid in drawing the reciprocal force diagram of a frame. Take as an example any case in which the lines of the force diagram cannot be drawn in succession without some artifice or auxiliary calculation or construction. (A case which may be taken is that of a King post or other roof truss, unsymmetrically loaded, for which the diagram is to be drawn without any preliminary calculation of the supporting forces.)
Prove that \[ \begin{vmatrix} \alpha^4-1 & \alpha^3 & \alpha \\ \beta^4-1 & \beta^3 & \beta \\ \gamma^4-1 & \gamma^3 & \gamma \end{vmatrix} = -(\beta-\gamma)(\gamma-\alpha)(\alpha-\beta)\alpha\beta\gamma\left(\beta\gamma+\gamma\alpha+\alpha\beta - \frac{1}{\beta\gamma} - \frac{1}{\gamma\alpha} - \frac{1}{\alpha\beta}\right). \]
An acute-angled isosceles triangular prism stands on a rough horizontal plane, and one of its side faces is subjected to increasing uniform normal pressure. Shew that equilibrium will be broken by sliding or tumbling as the angle of friction is less or greater than the vertical angle of the prism, supposed less than \(60^\circ\).
Solve the equations \begin{align*} x + y + z &= 5 \\ x^2 + y^2 + z^2 &= 13\frac{1}{2} \\ x^3 + y^3 + z^3 &= 44. \end{align*}
Obtain the tangential equation of the conic given by the general equation \[ ax^2+2hxy+by^2+2gx+2fy+c=0. \] Define conjugate points and conjugate lines with regard to a conic, and obtain the condition (1) that two points be conjugate, (2) that two lines be conjugate. Through two given points \(P\) and \(Q\) pairs of conjugate lines are drawn; prove that the locus of their point of intersection is a conic through \(P\) and \(Q\). Discuss the case when \(P\) and \(Q\) are conjugate points.