Exam practice: transformations, areas and volumes Problem set B

The graph of \(y=-x^3\) is transformed onto the graph of \(y=9-0.3x^3\) 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 \((0.01,\ 0.1)\).

1. (a) Find \(\frac{dy}{dx}\).
2. (b) Hence show that the equation of the tangent at this point is \(y=5x+0.05\).

The point P has coordinates \((1.2,\ 0.8)\). It is transformed by a stretch with scale factor \(\frac{1}{4}\) in the x-direction, then a stretch with scale factor \(5\) in the y-direction, then a translation of 2 units to the right and 3 units down.

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 glasshouse has an outer roof in the shape of a hemisphere of diameter 20 m, standing on vertical walls of height 6 m. Inside, there is a ceiling of glass panels shaped as a dome. The graph of \(y=-x^3\) is transformed onto the graph of \(y=14-0.06x^3\) 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 \(y=14-0.06x^3\) through 360° about the y-axis, between y = 0 and y = 14. The units are metres. Find the volume of the space between the outer shell and the ceiling.

The shape of a ceramic tile in a mosaic 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 \(0\le x\le 0.0625\), and \(f(x)=2x+0.125\) for \(0.0625<x\le 0.25\). 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 \(\frac{1}{2}\) in the y-direction, followed by a translation of 0.05 units to the right. Point A on f has coordinates \((0.25,\ 0.625)\), and B is the image of A.

1. (a) Show that the tangent to \(y=\sqrt{x}\) at the point \((0.0625,\ 0.25)\) is \(y=2x+0.125\).
2. (b) Find the coordinates of B, which are (p, q).
3. (c) The function g is given by \(g(x)=h(x)\) for \(0.05\le x\le a\) and \(g(x)=2x+b\) 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 \(x=0.25\).
5. (e) The area enclosed by the graph of g, the x-axis and the line x = p is 0.0231120 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.

A swimming-pool hall has an outer shell in the shape of a hemisphere of radius 9 m, standing on vertical walls of height 7 m. The inner ceiling is a dome.

1. (a) The ceiling is modelled by rotating the curve \(y=12-0.2x^2\) through 360° about the y-axis, between y = 0 and y = 12. The units are metres. Find the volume of the space between the outer shell and the ceiling.
2. (b) The space is filled with insulation costing $26 per cubic metre. Find the cost, to the nearest dollar.

A design for a garden path edging is drawn using a piecewise function. Units are metres. The function f is defined by \(f(x)=x^2\) for \(0\le x\le 2\) and \(f(x)=4x-4\) for \(2<x\le 3\). The point A has coordinates \((3,\ 8)\). 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 \(4\) in the y-direction, then a translation of 2 units to the left. 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.