Tag: spaces and boundaries - 2

Questions Related to spaces and boundaries - 2

The area of a rectangle of length (x + 2) units and breadth (x -8) units is

maths spaces and boundaries - 2 area of rectangular paths comparing areas problems on areas

  1. $x^2-16$

  2. $x^2-6x-16$

  3. $x^2-10x+16$

  4. $x^2-10$

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Correct Option: B
Explanation:

$l\times b=(x+2)(x-8)$
$=x^2+[(2)+(-8)]x+(2)(-8)$
$=x^2-6x-16$

A closed box made of steel of uniform thickness has length breadth and height 12 dm, 10 sm and 8 dm respectively If the thickness of the steel sheet is 1 dm then the inner surface area is

maths spaces and boundaries - 2 area of rectangular paths comparing areas problems on areas

  1. $ \displaystyle 456$ $\displaystyle dm^{2}$

  2. $ \displaystyle 376$ $\displaystyle dm^{2}$

  3. $ \displaystyle 264$ $\displaystyle dm^{2}$

  4. $ \displaystyle 696$ $\displaystyle dm^{2}$

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Correct Option: B
Explanation:

A closed box made by steel with uniform thickness length 12 dm ,breath  10 dm and 8 dm and thickness  of steel is 1 dm

Then inner length =12-2=10 dm ,breadth=10-2=8 dm and height=8-2=6 dm 
So surface area of box=$2(lw+wh+lh)=2(10\times 8)+(8\times 6)+(10\times 6)=2(80+48+60)=2\times 188=376 dm ^{2}$

The side of a square is 2 cm Semicircles are constructed on two sides of the square then the area of the whole figure is

maths how many squares area of rectangular paths comparing areas spaces and boundaries - 2

  1. $ \displaystyle (4+\pi )cm^{2} $

  2. $ \displaystyle (4+4\pi )cm^{2} $

  3. $ \displaystyle 4\pi cm^{2} $

  4. $ \displaystyle 8\pi cm^{2} $

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Correct Option: A
Explanation:

The side of square is 2 cm 

Then area of square =$(2)^{2}=4 cm^{2}$
The Semicircle constructed on two side diameter 2 cm then radius =1 cm
Then area of one semicircle =$\frac{\pi r^{2}}{2}=\frac{\pi (1)^{2}}{2}=\frac{\pi }{2}$
Then  area of two semicircle=$2\times \frac{\pi }{2}=\pi $
So total area of whole figure=$(4+\pi )cm^{2}$

A square of side x is taken A rectangle is cut out from this square such that one side of the rectangle is half that of the square and the other is $\displaystyle \frac{1}{3}$ of the first side of the rectangle
What is the area of the remaining portion?

maths how many squares area of rectangular paths comparing areas spaces and boundaries - 2

  1. $\displaystyle \left ( \frac{3}{4} \right )x^{2}$

  2. $\displaystyle \left ( \frac{7}{8} \right )x^{2}$

  3. $\displaystyle \left ( \frac{11}{12} \right )x^{2}$

  4. $\displaystyle \left ( \frac{15}{16} \right )x^{2}$

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Correct Option: C
Explanation:

Let the side of square is x 

Then area of square=$x^{2}$
Given one side of rectangle is half of square=$\frac{x}{2}$
And second side is $\frac{1}{3}$ of other side=$\frac{1}{3}\times \frac{x}{2}=\frac{x}{6}$
Then area of rectangle =$\frac{x}{2}\times \frac{x}{6}=\frac{(x)^{2}}{12}$
So remaining area of square =$x^{2}-\frac{(x)^{2}}{12}=\frac{11x^{2}}{12}$

Find the area of a triangle whose sides are 9 cm, 12 cm, and 15 cm.

maths how many squares area of rectangular paths comparing areas spaces and boundaries - 2

  1. $34cm^2$

  2. $44cm^2$

  3. $54cm^2$

  4. $55cm^2$

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Correct Option: C
Explanation:
Given sides of the triangle
$a=9$cm
$b=12$cm
$c=15$cm
The given triangle is a right angle triangle
because $9^2+12^2=15^2$
$\therefore$Area $=\dfrac{1}{2}\times 9\times 12=54$ sq cm
Area $=54$ sq cm.

Area bounded by $y|y|-x|x|=1,\ y|y|+x|x|=1$ and $y=|x|$ is

maths how many squares area of rectangular paths comparing areas spaces and boundaries - 2

  1. $\dfrac{\pi}{2}$

  2. ${\pi}$

  3. $\dfrac{\pi}{4}$

  4. $None\ of\ these$

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Correct Option: A

A farmer wishes to cultivate four different crops in equal areas in his $1$ car of land.. The land he should allocate for each crop is:  

maths how many squares area of rectangular paths comparing areas spaces and boundaries - 2

  1. $10\ m^{2}$

  2. $25\ m^{2}$

  3. $75\ m^{2}$

  4. $100\ m^{2}$

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Correct Option: A

A room is x meters long, y meters broad, and z meters high; find how many square meters of carpet will be required for the floor and how many square meters of paper for the walls?

maths how many squares area of rectangular paths comparing areas spaces and boundaries - 2

  1. xy and 2xy + 2yz

  2. xy and 2xz + 2yz

  3. xy and 2xy

  4. xy and 2xy + 2xz

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Correct Option: B
Explanation:

$Length = x, breadth = y$
Area of the floor $= xy m^2$
Area of four walls $= 2h(l + b) = 2z (x +y) = 2zx + 2yz$

Find the area of a square inscribed in a circle of radius $\displaystyle 5\sqrt{2}$ cm (in $\displaystyle cm^{2}$)

maths how many squares area of rectangular paths comparing areas spaces and boundaries - 2

  1. 75

  2. 100

  3. 125

  4. 150

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Correct Option: B
Explanation:

The diagonal of the square will be equal to the diameter of the circle.
So,diagonal of the square $ = 2 \times 5 \sqrt {2} = 10 \sqrt {2} $

Diagonal of a square $ = \sqrt {2} \times side $
So, $ 10 \sqrt {2} = \sqrt {2} \times side $
$ => Side  =  10  cm $

Area of the square $ = { side }^{ 2 } = { 10 }^{ 2 } = 100 $ sq cm

What is the maximum area of a four-sided plane with a perimeter of $12$ inches?

maths how many squares area of rectangular paths comparing areas spaces and boundaries - 2

  1. $8$ square inches

  2. $10$ square inches

  3. $6$ square inches

  4. $9$ square inches

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Correct Option: D
Explanation:

The area is maximum when the four-sided plane is a square.

The perimeter of a square $ = 4 \times$ side $= 12 $ inches
$ \Rightarrow$ side $= 3  $ inches

Now, area of the square $ =$ side $\times$ side $= 3 \times 3 = 9 $ square inches.