Exam practice: transformations, areas and volumes Problem set A

The graph of \(y=-x^3\) is transformed onto the graph of \(y=15-0.2x^3\) by a vertical translation of a units and a stretch parallel to the x-axis with scale factor b.

1. (a) Write down the value of a.
2. (b) Find the value of b.

The curve \(y=\sqrt{x}\) passes through the point \((0.04,\ 0.2)\).

1. (a) Find \(\frac{dy}{dx}\).
2. (b) Hence show that the equation of the tangent at this point is \(y=2.5x+0.1\).

The point P has coordinates \((0.9,\ 1.8)\). It is transformed by a stretch with scale factor \(3\) in the x-direction, then a stretch with scale factor \(\frac{1}{2}\) in the y-direction, then a translation of 1 unit to the left and 2 units up.

1. (a) Find the coordinates of the image after the two stretches.
2. (b) Find the coordinates of the image after the translation as well.

A planetarium has an outer shell in the shape of a hemisphere of diameter 30 m, standing on vertical walls of height 9 m. Inside, the ceiling is a dome. The graph of \(y=-x^3\) is transformed onto the graph of \(y=22-0.04x^3\) by a vertical translation of a units and a stretch parallel to the x-axis with scale factor b.

1. (a) Write down the value of a.
2. (b) Find the value of b.
3. (c) The ceiling is modelled by rotating the curve \(y=22-0.04x^3\) through 360° about the y-axis, between y = 0 and y = 22. The units are metres. Find the volume of the space between the outer shell and the ceiling.

The shape of a decorative stainless-steel panel is modelled by the region bounded by the graphs of f and g, the x-axis and the line segment [AB]. Units are metres. The function f is defined by \(f(x)=\sqrt{x}\) for \(0\le x\le 0.64\), and \(f(x)=0.625x+0.4\) for \(0.64<x\le 1.2\). The graph of g is obtained from the graph of f by a stretch with scale factor \(\frac{1}{2}\) in the x-direction, followed by a stretch with scale factor \(\frac{4}{5}\) in the y-direction, followed by a translation of 0.3 units to the right. Point A on f has coordinates \((1.2,\ 1.15)\), and B is the image of A.

1. (a) Show that the tangent to \(y=\sqrt{x}\) at the point \((0.64,\ 0.8)\) is \(y=0.625x+0.4\).
2. (b) Find the coordinates of B, which are (p, q).
3. (c) The function g is given by \(g(x)=h(x)\) for \(0.3\le x\le a\) and \(g(x)=x+b\) for \(a<x\le p\). Find an expression for h(x), the value of a and the value of b.
4. (d) Find the area enclosed by the graph of f, the x-axis and the line \(x=1.2\).
5. (e) The area enclosed by the graph of g, the x-axis and the line x = p is 0.354933 m² correct to six significant figures. Find the area of the region bounded by f, g, the x-axis and [AB], that is, the area under f minus the area under g and minus the area below AB.

The roof of a sports arena has an outer shell in the shape of a hemisphere of radius 14 m, standing on vertical walls of height 10 m. The inner ceiling is a dome.

1. (a) The ceiling is modelled by rotating the curve \(y=20-0.2x^2\) through 360° about the y-axis, between y = 0 and y = 20. The units are metres. Find the volume of the space between the outer shell and the ceiling.
2. (b) The space is filled with foam costing $34 per cubic metre. Find the cost, to the nearest dollar.

A pattern for a sail is drawn using a piecewise function. Units are metres. The function f is defined by \(f(x)=x^2\) for \(0\le x\le 1\) and \(f(x)=2x-1\) for \(1<x\le 3\). The point A has coordinates \((3,\ 5)\). The graph of g is obtained from the graph of f by a stretch with scale factor \(2\) in the x-direction, then a stretch with scale factor \(3\) in the y-direction, then a translation of 1 units to the right. B is the image of A.

1. (a) Find the coordinates of B.
2. (b) Write down g(x) on each of its two pieces, with the interval for each.
3. (c) Find the area enclosed by the graph of f, the x-axis and the line \(x=3\).
4. (d) Hence find the area enclosed by the graph of g and the x-axis.