Question
The straight metal arm of a windscreen wiper on a car rotates in a circular motion from a
pivot point, O, through an angle of 140°. The windscreen is cleared by a rubber blade of
length 46 cm that is attached to the metal arm between points A and B. The total length
of the metal arm, OB, is 56cm.
The part of the windscreen cleared by the rubber blade is shown unshaded in the
following diagram.
(a) Calculate the length of the arc made by B, the end of the rubber blade.
(b) Determine the area of the windscreen that is cleared by the rubber blade.
Answer/Explanation
Ans:
(a) attempt to substitute into length of arc formula
\(\frac{140^0}{360^0} \times 2 \pi \times 56\)
137 cm \((136.833…., \frac{392 \pi}{9} cm)\)
(b) subtracting two substituted area of sectors formulae
\((\frac{140^0}{360^0} \times \pi 56^2)\) – \((\frac{140^0}{360^0} \times \pi \times 10^2)\) OR \(\frac{140^0}{360^0} \times \pi \times (56^2 – 10^2)\)
3710 \(cm^2\) (3709.17… \(cm^2\))
Question
Three towns, A, B and C are represented as coordinates on a map, where the x and y axes
represent the distances east and north of an origin, respectively, measured in kilometres.
Town A is located at (−6, −1) and town B is located at (8 , 6). A road runs along the
perpendicular bisector of [AB]. This information is shown in the following diagram.
(a) Find the equation of the line that the road follows.
Town C is due north of town A and the road passes through town C.
(b) Find the y-coordinate of town C.
Answer/Explanation
Ans:
(a) midpoint (1, 2.5)
\(m_{AB}=\frac{6-(-1)}{8-(-6)} = \frac{1}{2}\)
\(m_{\perp} = -2\)
y – 2.5 = -2 (x – 1) OR \(y=-2x + \frac{9}{2}\) OR 4x + 2y – 9 = 0
(b) substituting x = -6 into equation from part (a)
\(y = -2(-6) + \frac{9}{2}\)
y = 16.5