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Bài tập CALCULUS 74
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Created by T. Madas
Question 252
(****+)
A
D
F
x
C
B
E
The figure above shows the design of a horse feeder which in the shape of a hollow,
open topped triangular prism.
The triangular faces at the two ends of the feeder are isosceles and right angled, so that
AB = BC = DE = EF and ABC = DEF = 90° .
The triangular faces are vertical, and the edges AD , BE and CF are horizontal.
The capacity of the feeder is 4 m3 .
a) Show that the surface area, A m 2 , of the feeder is given by
A=
1 2 16 2
x +
,
2
x
where x is the length of AC .
b) Determine by differentiation the value of x for which A is stationary, giving
the answer in the form k 2 , where k is an integer.
c) Show that the value of x found in part (b) gives the minimum value for A .
[continues overleaf]
Created by T. Madas
Created by T. Madas
[continued from overleaf]
d) Show, by exact calculations, that the minimum surface area of the feeder is
12 m 2 .
e) Show further that the length ED is equal to the length EB .
C2V , x = 2 2 ≈ 2.82 , ED = EB = 2
Created by T. Madas
Created by T. Madas
Question 253
(****+)
y
M
y = − x2 + 8x − 7
A ( 6,5 )
x
O
The figure above shows the quadratic curve with equation
y = − x2 + 8x − 7 .
The point M is the maximum point of the curve and A is another point on the curve
whose coordinates are ( 6,5 ) .
Find the exact area of the shaded region, bounded by the curve, the x axis and the
straight line segment from A to M .
SYN-M , area = 104
3
Created by T. Madas
