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Arithmetic Aptitude :: Volume and Surface Area

1. Given 3 lines in the plane such that the points of intersection form a triangle with sides of length 20, 20 and 30, the number of points equidistant from all the 3 lines is

Answer: 0

Explanation:

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2. A sphere and a cube have the same surface area. Find the ratio of their volumes?

Answer: √6 : √π

Explanation:

SA of a sphere: 4πr²
SA of a cube: 6x²

4πr² = 6x²
r²/x² = 6 / 4π
(r/x)² = 3 / 2π
r/x = (3 / 2π)^0.5

Volume of a sphere: 4/3 πr³
Volume of a cube: x³
Find the ratio meaning (4/3 πr³)/x³
= (4π/3)(r³/x³)
= (4π/3)(r/x)³
= (4π/3)(3 / 2π)^1.5
= (2²π/3)[3^1.5 / (2^1.5)(π^1.5)]
= √2√3 / √π
= √(6/π)

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3. The slant height of a right circular cone is 10 m and its height is 8 m. Find the area of its curved surface.

Answer: 60π m2

Explanation:

l = 10m
h = 8m
so r = √(l^2 - h^2) = √(10^2 - 8^2) = 6m
So Curved surface area = (π * r * l) = ( π * 6 * 10) m2 = 60π m2.

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4. A builder has to pour a concrete slab 12 centimeters thick to cover an area 10 meters long and 2 meters wide. How many cubic meters of concrete will the builder need?

Answer: 2.4

Explanation:

Thikness= 12cm. =0.12m.
area is 10m. long and 2m. wide;
So, total concrete needed is = the volume of the area = (0.12 * 10 * 2)= 2.4 cubic meter.

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5. Find the slant height, volume, curved surface area and the whole surface area of a cone of radius 21 cm and height 28 cm.

Answer: 3696 cm^2

Explanation:

Slant Height, l = ?(r^2 + h^2) =?(21^2 + 28^2) = ?1225 = 35 cm
Volume = 1/3?r^2h = (1/3 * 22/7 * 21 * 21 * 28) cm^3 = 12936 cm^3
Curved surface area = ?rl = 22/7 * 21 *35 cm^3 = 2310 cm^2
So,total Surface Area = (?rl + ?r^2) = (2310 + 22/7 * 21 * 21) cm^2 = 3696 cm^2

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6. If the wheel of a bicycle makes 560 revolutions in travelling 1.1 km, what is its radius?

Answer:

Explanation:

The distance covered by the wheel in 560 revolutions = 1100 m .
Hence, the distance covered per revolution = 1100/560 = 55/28 metres.
The distance covered in one revolution = circumference of the wheel.
Circumference = (2 * pi * r)
=> 55/28 = 2 * 22/7 * r
=> r = 31.25 cm.

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7. If each edge of a cube is increased by 50%, find the percentage increase in its surface area.

Answer: 125%

Explanation:

Let the original length of each edge = a
Then, Original surface area = 6a^2
New surface area = 6 * (3a/2)^2 = 27a^2/2
Increase percent in surface area = (15/2a^2 * 1/(6a^2) * 100)% = 125%

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8. If the radius of the sphere is increased by 50%, find the increase percent in volume and the increase percent in the surface area.

Answer: 125%

Explanation:

Let the original radius = R. Then, new radius = 150/100 R = 3R/2
Original Volume = 4/3?R^3, New volume = 4/3?(3 R/2)^3 = 9?R^3/2
Original surface area = 4?R^2 , New surface area = 4?(3R/2)^2 = 9?R^2
Increase % in surface area = (5?R^2/4?R^2 * 100)% = 125%

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9. A cylindrical container has a radius of eight inches with a height of three inches. Compute how many inches should be added to either the radius or height to give the same increase in volume?

Answer: 16/3 inches

Explanation:

Let x be the amount of increase.
The volume will increase by the same amount if the radius increased or the height is increased.
So, the effect on increasing height is equal to the effect on increasing the radius.
=> (22/7)*8*8*(3+x) = (22/7)*(8+x)*(8+x)*3
Solving the quadratic equation
we get the x = 0 or 16/3.
So, the possible increase would be by 16/3 inches.

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10. A circular tent is cylindrical to a height of 3 meters and conical above it. If its diameter is 105 m and the slant height of conical portion is 53m, calculate the length of the canvas 5 m wide to make the required tent?

Answer: 1947

Explanation:

Surface area of Tent= surface area of the cylinder + surface area of the cone
=(2*pi*r*h)+ (pi*r*l)
=2*(22/7)*(105/2)*3+(22/7)*(105/2)*53
=9735 sq.m
l*b=9735
b=5m,
l=9735/5 => l=1947m

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