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Big Questions Practice
1. Clara organizes cans in triangular piles, where each row has one less can than the rowbelow. For example, the pile of 15 cans shown has 5 cans in the bottom row and 4cans in the row above it.
(a ! pile has "# cans in the bottom row. $how that the pile contains "1# cans.
(b %here are &"4# cans in a pile. 'ow man cans are in the bottom row)
(c (i %here are Scans and the are organized in a triangular pile with ncans in
the bottom row. $how that n"* n+ "S #.
(ii Clara has "1## cans. -xplain wh she cannot organize them in a triangular pile.
2. et Snbe the sum of the first nterms of the arithmetic series " * 4 * / * 0.
(a Find
(i S4 (ii S1##.
etM
1#
"1.
(b (i FindM". (ii $how thatM
&
1#
/1.
2t ma now be assumed thatMn
1#
"1 n, for n4. %he sum Tnis defined b
TnM1*M
"*M
&* ... *M
n.
(c (i 3rite downM4. (ii Find T4.
(d sing our results from part (a (ii, find T1##.
&. ! cit is concerned about pollution, and decides to loo at the number of people using taxis. !t the end of theear "###, there were "6# taxis in the cit. !fter nears the number of taxis, T, in the cit is given b
T "6# 1.1"n.
(a (i Find the number of taxis in the cit at the end of "##5.
(ii Find the ear in which the number of taxis is double the number of taxis there were at theend of "###.
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(b !t the end of "### there were "5 /## people in the cit who used taxis. !fter nears the numberof people,P, in the cit who used taxis is given b
P .e7#1#
###5/#"#.18 n+
(i Find the value ofPat the end of "##5, giving our answer to the nearest whole number.
(ii !fter seven complete ears, will the value ofPbe double its value at the end of "###)9ustif our answer.
(c etRbe the ratio of the number of people using taxis in the cit to the number of taxis. %he cit
will reduce the number of taxis ifR (x. etAbe the area of the region enclosed b the graph of g and thex=axis, between
x # andx a, where a>#. ?iven thatA ", find the value of a.
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5. ! Ferris wheel with centre @ and a radius of 15 metres is represented in the diagram below. 2nitiall seat ! is
at ground level. %he next seat is A, where A@B! /
C.
(a Find the length of the arc !A.
(b Find the area of the sector !@A.
(c %he wheel turns clocwise through an angle
of&
". Find the height of ! above the
ground.
%he height, hmetres, of seat C above the groundafter tminutes, can be modelled b the function
h(t 15 + 15 cos
+
4C"t .
(d (i Find the height of seat C when t4
C.
(ii Find the initial height of seat C.
(iii Find the time at which seat C first reaches its highest point.
(e Find h> (t.
(f For # t,
(i setch the graph of h>
(ii find the time at which the height is changing most rapidl.
/. %he following is the cumulative fre
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1 5 #
1 4 #
1 & #
1 " #
1 1 #
1 # #
7 #
6 #
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/ #
5 #
4 #
& #
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1 #
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