A farmer wanted to make a trough for cows to drink water. He had a metal sheet measuring
240cm by 120cm and 1cm thick. The density of the metal is 2.5g/cm
3
. A square of sides 30cm
is removed from each corner of the rectangle and the remaining part folded to form an open
cuboid.
(a) Sketch the sheet after removing the squares for the four corners, showing all the dimensions
(b) Calculate:-
(i) The area of the metal which forms the cuboid
(ii) The mass of the empty cuboid in Kilograms
(b) The cuboid is filled with water whose density is 1g/cm
3
. Calculate the mass of the
cuboid
when full
of water
12.
A rectangular sheet of cardboard is 8cm long and 5cm wide. Equal squares are cut away at
each corner and the remainder is folded so as to form an open box. Find the maximum volume
13.
(a) Find the equation of the normal to the curve :- y = x
3
– 2x – 1 at ( 1, -2)
(b) Determine the nature of the turning points to the curve y = x
3
– 3x + 2; Hence in the
space provided below, sketch the curve
14.
A particle moves in a straight line so that its velocity, v/m/s at time t seconds where
t
0
is given by
v = 28 + t – 2t
2
Find:-
(a) The time when
P
is instantaneously at rest
(b) The speed of
P
at the instant when the acceleration of
P
is zero
(c) Given that
P
passes through the point
O
of the line when t = 0;
(i) Find the distance of
P
from
O
when
P
is instantaneously at rest
15.
A particle moves such that
t
seconds after passing a given point
O
, its distance
S
metres
from
O
is given by
S= t (t-2) (t-1)
(a) Find its velocity when
t
= 2 seconds
(b) Find its minimum velocity

(c)Find the time when the particle is momentarily at rest
(d) Find its acceleration when
t
= 3seconds

64.Approximation of area
1
Use trapezoidal rule to estimate the area bounded by the curve
y = 8 + 2x – x
2
for -1
x
3
using 5 ordinates
2.
(a) Using trapezoidal rule, estimate the area under the curve y = ½ x
2
– 2 between x = 2
and x = 8
and x-axis. Use six strips
(b) (i) Use integration to evaluate the exact area under the curve
(ii) Find the percentage error in calculating the area using trapezoidal rule
3
(a) Using trapezoidal rule, estimate the area under the curve y = ½ x
2
– 2 between x = 2
and x = 8
and x-axis. Use six strips
(b) (i) Use integration to evaluate the exact area under the curve
(ii) Find the percentage error in calculating the area using trapezoidal rule
4
The figure below shows the graphs of y = 2x + 3 and y = -2x
2
+ 3x + 4
(a) determine the co-ordinates of Q, the intersection of the two graphs
(b) Find the exact area of the shaded region
5.
The table below shows some values of the function; y =
x
2
+ 2
x
– 3 for -6
≤
x
≤
-3
x
-6
-5.75
-5.5
-5.25
-5
-4.75
-4.5
-4.25
-4.0
-3.75
-3.5
-3.25
-3
y
21
18.56
14.06
10.06
8.25
5
2.25
1.06
0
(a) complete the table
(b) using the completed table and the mid-ordinate rule with six ordinates, estimate the
area of the region bounded by the curve; y =
x
2
+ 2
x
– 3 and the lines y = 0 ,
x
= -6 and
x
= -3
(c) (i) by integration find the actual are of the region in
(b)
above
(ii) Calculate the percentage error arising from the estimate in
(b)
6
Complete the table below for y = 5x
2
– 2x + 2. Estimate the area bounded by the curve, the
x – axis, the lines x = 2 and x = 7 using the trapezoidal rule with strips of unit length.
x
2
2.5
3
3.5
4
4.5
5
5.5
6
6.5
7
y
18
56.25
74
117
200.25
6. Integration
1.

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