Problem 1 :
An open box is to be made from a square piece of material, 24 cm on a side, by cutting equal squares from the corners and turning up the sides as shown. Express the volume V of the box as a function of x.
Solution :
Since the original shape is square, length of all sides will be equal.
length = width = 24 - 2x
height = x
Volume of cuboid = length ⋅ width ⋅ height
= (24 - 2x) ⋅ (24 - 2x) ⋅ x
= x(24 - 2x)2
= x [242 - 2(24) (2x) + (2x)2]
= x [576 - 96x + 4x2]
V (x) = 4x3 - 96x2 + 576x
Hence the volume of the cuboid is 4x3 - 96x2 + 576x.
Problem 2 :
A function f is defined by f (x) = 3−2x . Find x such that f (x2) = (f (x))2 .
Solution :
Given that :
f (x) = 3 − 2x
f (x2) = 3 − 2x2 ------(1)
(f (x))2 = (3 − 2x)2
= 32 - 2(3)(2x) + (2x)2
(f (x))2 = 9 - 12x + 4x2 ----(2)
(1) = (2)
3 − 2x2 = 9 - 12x + 4x2
4x2 + 2x2 - 12x + 9 - 3 = 0
6x2 - 12x + 6 = 0
x2 - 2x + 1 = 0
(x - 1)2 = 0
x - 1 = 0 (or) x - 1 = 0
x = 1 (or) x = 1
Problem 3 :
A plane is flying at a speed of 500 km per hour. Express the distance d travelled by the plane as function of time t in hours.
Solution :
Time = Distance / speed
Speed of plane = 500 km per hour
Distance travelled = d
time taken = t in hours
t = d/500
d = 500t
Hence the required distance is 500 t.
Problem 4 :
The data in the adjacent table depicts the length of a woman’s forehand and her corresponding height. Based on this data, a student finds a relationship between the height (y) and the forehand length(x) as y = ax +b , where a, b are constants.
(i) Check if this relation is a function.
(ii) Find a and b.
(iii) Find the height of a woman whose forehand length is 40 cm.
(iv) Find the length of forehand of a woman if her height is 53.3 inches.
Solution :
y = ax + b
(i) For every values of x, we get different values of y.Hence it is a function.
(ii)
y = a x + b x = 45.5, y = 65.5 65.5 = a(45.5) + b 65.5 = 45.5 a + b --(1) |
y = a x + b x = 35, y = 56 56 = a(35) + b 56 = 35 a + b --(2) |
By solving these two equation, we get a and b
(1) - (2)
(45.5 a + b) - (35a + b) = 65.5 - 56
45.5 a - 35a + b - b = 9.5
10.5a = 9.5
a = 9.5/10.5
a = 0.90
Substitute a = 0.90 in (1),
45.5(0.90) + b = 65.5
b = 65.5 - 40.95
b = 24.5
y = 0.9x + 24.5
(iii) Find the height of a woman whose forehand length is 40 cm.
y = ? if x = 40
y = 0.9(40) + 24.5
y = 60.5
Height of woman is 60.5 inches.
(iv) x = ? if y = 53.3
53.3 = 0.9x + 24.5
53.3 - 24.5 = 0.9x
28.8/0.9 = x
x = 32
Length of forehand is 32 inches.
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