A taxi firm's roof sign is a rectangle of length \(3x\) cm and width \(x\) cm, with a semicircle of diameter \(x\) cm attached to one short end, as shown. T...
Question 1 Report
A taxi firm's roof sign is a rectangle of length \(3x\) cm and width \(x\) cm, with a semicircle of diameter \(x\) cm attached to one short end, as shown. The perimeter of the sign is \(58\) cm.
- Show that the perimeter is given by \(x\left(7 + \dfrac{\pi}{2}\right)\) cm. (2)
- Form and solve an equation to find \(x\), giving your answer correct to 3 significant figures. (2)
Answer Details
This "show that" question requires the perimeter to be built up edge by edge in terms of \( x \), and then factorised into the given form.
(a) The sign's outline consists of: the long top side, \( 3x \); the long bottom side, \( 3x \); the short left side, \( x \); and, in place of the short right side, a semicircular arc of diameter \( x \), which has length \( \pi \times \dfrac{x}{2} \). Adding these: \( 3x + 3x + x + \dfrac{\pi x}{2} = 7x + \dfrac{\pi x}{2} \) [1 mark]. Factoring out \( x \) gives \( x\left(7 + \dfrac{\pi}{2}\right) \) cm, as required [1 mark].
(b) Setting the perimeter expression equal to 58 forms the equation \( x\left(7 + \dfrac{\pi}{2}\right) = 58 \) [1 mark]. Dividing both sides by \( \left(7 + \dfrac{\pi}{2}\right) = 8.5708\ldots \) gives \( x = \dfrac{58}{8.5708\ldots} = 6.77 \) to 3 significant figures [1 mark].
The straight edge opposite the semicircle is not included in the perimeter, since the semicircular arc replaces it; only three straight sides plus one curved arc bound this shape.
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