This document contains a mathematics project with multiple parts and questions. 1) Part 1 involves solving simultaneous equations to find the maximum area of a fence given certain constraints. The maximum area is 1250 square meters. 2) Part 2 involves using a volume equation and given values to find the largest possible volume, which is 2000 cubic centimeters. 3) Part 3 involves solving word problems related to time, population, and trigonometric functions.

1. JABATAN PENDIDIKAN SELANGOR
KERJA PROJEK MATEMATIK TAMBAHAN 2015
RANGKA PENYELESAIAN TUGASAN
PART 1
(a) Accept any relevant answer
(b) Accept any relevant answer
PART 2
(a)
Length of fence = 4x + 2y = 200 .................
Area ,A = xy ........................
From
2x + y = 100
y = 100 – 2x ........
Substitute into
A = x( 100 – 2x)
= 100x – 2x2
x
dx
dA
4100 
25
041000,maximumFor


x
x,
dx
dA
y = 100 – 2 (25) = 50
42
2

xd
Ad
Therefore,maximum area = 25 X 50
= 1250m2
x x
y
x x
y
1
2
2
1
3
3

2.     
 
155
090024012
90024012
9001204
230230
2
2
23
orh
hh
hhh'V
hhh
hhhhV
hWIV)b






V(5)=2000
V(15)=0
Largest possible volume = 2000 cm3
PART 3
(a)(i)
(ii) 3.30 pm @ 1530
(iii) 3600 people
(iv) 25701800)
6
cos(1800 
t
4278.0)
6
cos( t

 67.64
 67.244,33.115
6
t

t = 3.844 , 8.156
1 2 3 4 5 6 7 8 9 10 11 12 13
3500
3000
2500
2000
1500
1000
500
-500
f x  = -1800cos
x
6 +1800
t
P( t)

3. FURTHER EXPLORATION
(a) Accept any relevant answer
(b) (i) (a) x : number of model x cabinets purchased
y : number of model y cabinets purchased
𝑥 ≥ 0, 𝑦 ≥ 0
Cost : 100𝑥 + 200𝑦 ≤ 1400 𝑜𝑟 𝑦 ≤ −
1
2
𝑥 + 7
Space : 0.6𝑥 + 0.8𝑦 ≤ 7.2 𝑜𝑟 𝑦 ≤ −
3
4
𝑥 + 9
No. of cabinets: 𝑦 ≤
3
2
𝑥
Volume: 𝑣 = 0.8𝑥 + 1.2𝑦
(b) Refer to graph
(ii) Method (a):
Using the optimum linear line 𝑣 = 0.8𝑥 + 1.2𝑦. Maximum point (8,3)
Maximum volume =0.8(8)+1.2(3) = 10 cubic meters
Method (b):
Using the corner points
Points Volume (m3)
(4,5) 0.8(4) + 1.2(5)=9.2
(8,3) 0.8(8) + 1.2(3)=10.0
Maximum volume=10 cubic meters
(iii)
Combination (x,y) Volume (m3) Cost (RM)
(4,1) 4.4 600.00
(4,2) 5.6 800.00
(4,3) 6.8 1000.00
(4,4) 8 1200.00
(4,5) 9.2 1400.00
(5,1) 5.2 700.00
(5,2) 6.4 900.00
(5,3) 7.6 1100.00
(5,4) 8.8 1300.00
(6,1) 6 800.00
(6,2) 7.2 1000.00
(6,3) 8.4 1200.00
(6,4) 9.6 1400.00
(7,1) 6.8 900.00
(7,2) 8 1100.00
(7,3) 9.2 1300.00
(8,1) 7.6 1000.00
(8,2) 8.8 1200.00
(8,3) 10 1400.00
(9,1) 8.4 1100.00
(9,2) 9.6 1300.00

4. (iv) Accept any relevant answer.
0 2 4 6 8 10 12 14 x
Graph y against x
y
1
2
3
4
5
6
7
8
9
10
xy
2
3

9
4
3
 xy
7
2
2
 xy
(8,3)
(4,5)
y = 0.8x + 1.2y
REFLECTION: Accept any relevant answer