Similarity：練習問題

Two solids are mathematically similar. The ratio of their surface areas is \(9:25\). If the volume of the smaller solid is \(108 \text{ cm}^3\), calculate the volume of the larger solid.

Two similar cones have heights h and 3h. If the volume of the larger cone is \(108 \text{ cm}^3\), find the volume of the smaller cone.

In the diagram, a triangle ABC has sides of lengths \(AB = 6\) cm, \(BC = 8\) cm, and \(AC = 10\) cm. A similar triangle PQR has its longest side \(PR = 25\) cm. Calculate the length of the side PQ which corresponds to AB.

Two mathematically similar decorative statues, \(X\) and \(Y\), are made of the same material.
Statue \(X\) has a mass of \(54\text{ g}\) and a base area of \(36\text{ cm}^2\).
Statue \(Y\) has a mass of \(128\text{ g}\).

Calculate the base area of statue \(Y\).

Two mathematically similar geometric cones, \(A\) and \(B\), have surface areas of \(45\pi\text{ cm}^2\) and \(80\pi\text{ cm}^2\) respectively.
The height of cone \(A\) is \(18\text{ cm}\).

Calculate the height of cone \(B\).

Two solid metal spheres are melted down and recast into a single large sphere.
The radius of the first sphere is 3 cm and the radius of the second sphere is 4 cm.

(a) Show that the radius, R, of the new large sphere is \(\sqrt[3]{91}\) cm. [3]
(b) Find the ratio of the surface area of the smaller sphere to the surface area of the new large sphere. Give your answer in the form \(1 : n\). [2]

In a diagram, triangle ABC is similar to triangle ADE such that BC is parallel to DE.
The point B lies on AD and the point C lies on AE.
It is given that AB = x, BD = x + 3, BC = 4, and DE = x + 2.

(a) Form an equation in terms of x and show that it simplifies to \(x^2 - 6x - 12 = 0\). [3]
(b) Solve the equation \(x^2 - 6x - 12 = 0\) to find the value of x, giving your answer to 2 decimal places. [3]
(c) Using your value of x, calculate the length of AD. [1]