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Geometry and Mensuration: Level 2 Test 9

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Question 1
Suface area of a cuboid is 22 cm2 and the sum of the lengths of all its edges is 24 cm. Length of each diagonal of the cuboid (in cm) is
A
145
B
200
C
122
D
none
Question 1 Explanation: 
$ \displaystyle \begin{array}{l}Let\text{ }the\text{ }length\text{ },\text{ }width\text{ }and\text{ }height\text{ }be\text{ }l\text{ },\text{ }w\text{ }and\text{ }h\text{ }units\text{ }respectively\\l+b+w=\frac{24}{4}=6\\lb+bw+lw=\frac{22}{2}=11\\Now\,\,{{l}^{2}}+{{b}^{2}}+{{w}^{2}}+2lb+2bw+2lw={{(l+b+w)}^{2}}\\{{l}^{2}}+{{b}^{2}}+{{w}^{2}}+22=36\\{{l}^{2}}+{{b}^{2}}+{{w}^{2}}=14\\\sqrt{{{l}^{2}}+{{b}^{2}}+{{w}^{2}}}=\sqrt{14}\end{array}$
Question 2
A slab of ice 8 inches in length, 11 inches in breadth, and 2 inches thick was melted and re-solidified in the form of a rod of 8 inches diameter. The length of such a rod, in inches, in nearest to:
A
3
B
3·5
C
4
D
4·5
Question 2 Explanation: 
The volume of the ice slab =8X11X2
The volume of the rod (assuming length = h) =Πx42h = 8 x 11 x 2
h = 3.5
Correct option is (b)
Question 3
Four friends start from four towns, which are at the four comers of an imaginary rectangle. They meet at a point which falls inside the rectangle, after travelling the distances of 40 m, 50 m and 60 m. The maximum distance that the fourth could have travelled is approximately:
A
67 m
B
52 m
C
22.5 m
D
Cannot be determined
Question 3 Explanation: 
Now we know that if we assume the length required =p resp,
then 50 x 40 = 60 x p
=> p = 50 x 40/60
=> p = 66.67
The correct option is (a)
Question 4
The length of a ladder is exactly equal to the height of the wall it is leaning against. H lower end of the ladder kept on a stool of height 3 m and the stool is kept 9 m away from the wall, the upper end of the ladder coincides with the top of the wall. Then, the height of the wall is:
A
12 m
B
15 m
C
18 m
D
11 m
Question 4 Explanation: 
101
Let the length of the ladder be l m
l2 = (1-3)2 + 92
2l -3 = 27
l = 15 cm
Question 5
An equilateral triangle and a regular hexagon have equal perimeters. The ratio of the area of the triangle and that of the hexagon is
A
1: 1
B
2: 3
C
3: 2
D
3: 4
Question 5 Explanation: 
Let the individual sides of the hexagon be 1 cm
Thus the perimeter =6 and the side of the equilateral triangle is 2cm.
Thus the area of the equilateral triangle = √3 x 4 = √3 cm.
Thus the area of the hexagon= {(6 x √3)/4} x 12 = 3/2√3
Thus the ratio of the triangle to hexagon is 2:3.
Correct option is (b)
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