Question 1

Paper 1Hard15 marks

A workshop is producing identical circular badges for a display. Their common radius is measured as 3.43.4 cm, correct to the nearest 0.10.1 cm. The combined face area of a batch is the sum of the areas of its badges. The display requires a combined face area of at least 680 cm2680\text{ cm}^2

(a)

State the lower and upper bounds for the actual radius rr

[2]
(b)

Let A cm2A\text{ cm}^2 be the area of one badge. Show that

11.2225π≤A<11.9025π.11.2225\pi \leq A < 11.9025\pi.
[3]
(c)

Calculate the estimated area of one badge, based on the reported radius, giving your answer to 3 significant figures. Find a conservative upper bound for the percentage error in this estimate, using the estimated area as the denominator. Give the percentage bound to 3 significant figures, rounding upwards.

[4]
(d)

Using the estimated area from part (c), determine the minimum number of badges indicated by the estimate to meet the display’s area requirement.

[2]
(e)

The workshop orders the number of badges found in part (d). Assess the supervisor’s claim that this order is guaranteed to meet the display’s area requirement, and determine the smallest order that would guarantee it.

[4]

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