Activities
P.o.t.W. #6 Solution
(a) The overlapping region consists of two equal segments. Find the area of one of the segments and double it. Let \({\textrm{\theta }} be the central angle between two radii drawn to the two points...
Activities
P.o.t.W. #21 Solution
(a) \(r = \dfrac{{{u_2}}}{{{u_1}}} = \dfrac{{\;\;\dfrac{{x{{\left( {x + 1} \right)}^2}}}{{{{\left( {x - 2} \right)}^2}}}\;\;}}{{\dfrac{{x\left( {x + 1} \right)}}{{x - 2}}}} = \dfrac{{x{{\left( {x +...
Activities
P.o.t.W. #17 Solution
(a) \displaystyle\int {\sqrt {1 - {x^2}} } \,{\textrm{d}}x x = \sin {\theta }\;\;\; \Rightarrow \;\;\;{\textrm{d}}x = \cos {\theta }\,{\textrm{d\theta }} substituting: \displaystyle\int {\sqrt {1 -...
Activities
P.o.t.W. #12 Solution
(a) Firstly, wherever point C is located, angle A and angle B of triangle ABC must both be acute angles. Hence, angle C must be obtuse for triangle ABC to be an obtuse triangle. Secondly, the probability...
Activities
P.o.t.W. #8 Solution
Consider the right triangle OAT where O is the origin and T is the point where the circle is tangent to the graph of \(y = k\left| x \right| such that \(OA = a and \(AT = b. Because the slope of the right...
Activities
P.o.t.W. #25 Solution
■ No GDC ■ Given: \(a + b = 1 and \({a^4} + {b^4} = 7 Let \(P = {a^2} + {b^2} and \(Q = ab. \({\left( {a + b} \right)^2} = 1 and \({\left( {a + b} \right)^2} = {a^2} + {b^2} + 2ab; hence, \(P +...
Activities
P.o.t.W. #30 Solution
The number of handshakes is the number of combinations of 33 items chosen 2 at a time. {}_{33}{C_2} = 528 Hence, there are 528 handshakes. There are 12 handshakes occurring during each 10-second interval....
Activities
P.o.t.W. #15 Solution
Consider the calculations for the area of regular polygons with \(n = 3,\;4,\;5 and \(6 sides inscribed within a circle of radius \(r. In each case, the polygon can be divided into \(2n identical right...
Activities
P.o.t.W. #19 Solution
If two polygons are similar (i.e. corresponding sides are proportional and corresponding angles are equal), then the ratio of the areas of the two polygons is equal to the square of the ratio of a pair...
Activities
P.o.t.W. #35 Solution
The only results where both coins show the same number of heads is when each coin shows 0, 1, 2 or 3 heads. Calculate the probability of showing 0, 1, 2 or 3 heads for each coin. Coin A: \displaystyle\mathrm{P}\left(\mathrm{A}{=0}\right)={}_3{C_0}{\left({\frac12}\right)^3}=\frac18...