Tag: measures and the circle

Questions Related to measures and the circle

State whether true or false : 

If $s$ is the perimeter of the base of a prism, $n$ is the number of sides of the base, $S$ is the total length of the edges and $h$ is the height, then $S=nh+2s$.

  1. True

  2. False


Correct Option: A
Explanation:

True. Total length of Prism $= S$
Length of all the heights $= nh$
Perimeter of base and head $= 2s$
$\therefore S = nh + 2s$

The base of right prism is a triangle whose perimeter is 28 cm and the inradius of the triangle is 4 cm. If the volume of the prism is 366 cc, then its height is 

  1. 6.54 cm

  2. 8 cm

  3. 4 cm

  4. None of these


Correct Option: A
Explanation:

Perimeter of triangle is $2{s}=28\ cm\implies s=14$

Inradius of triangle $r=4\implies \Delta=r.s=56$
Volume of prism $=366$ cc
$\implies $ Area of triangle $\times $ height $=366$
Height $=\dfrac{366}{56}=6.54$

For a prism, $A = 60^o$, $\mu = \sqrt{\dfrac{7}{3}}$, then the minimum possible angle of incidence, so that the light ray is refracted from the second surface is

  1. $30^o$

  2. $60^o$

  3. $90^o$

  4. $40^o$


Correct Option: A

The slant height of a right pyramid having square base of side $10cm$ and vertical height $15cm$ is

  1. $5\sqrt{10}cm$

  2. $6\sqrt{10}cm$

  3. $7\sqrt{10}cm$

  4. $8\sqrt{10}cm$


Correct Option: A

Consider an incomplete pyramid of balls on a square base having $18$ players, and having $13$ balls on each side of the top layer. Then the total number $N$ of balls in that pyramid satisfies 

  1. $ 9000 < N <10000$

  2. $8000 < N < 9000$

  3. $7000 < N < 8000$

  4. $ 10000 < N < 12000 $


Correct Option: B
Explanation:

Top layer has $(13 \times 13)$ balls 
Similarly one layer below top layer will have $(14 \times 14) $ balls and we have $18$ lesens to total number of ball 
$ N = (13)^2 + (14)^2 + . . . . + (30)^2 $

$\displaystyle N = \frac {30 \times 31 \times 61} {6} = \frac {12 \times 13 \times 25} {6}$

$ N = 8805 $

The circumference of a 1 cm thick pipe is 44 cm. The level of water that 7 cm of pipe can hold is

  1. $798 cm^3$

  2. $308 cm^3$

  3. $792 cm^3$

  4. $795 cm^3$


Correct Option: C
Explanation:

Given that

$2\pi r=44$
$r=7\ cm$

So, the inner radius of the pipe $=7-1=6\ cm$

Therefore, the volume of the pipe
$=\pi r^2h$
$=\pi \times 6^2\times 7$
$=792\ cm^3$

Hence, this is the answer.

The base of a right prism is a square of perimeter 20 cm and its height is 30 cm. The volume of the prism is

  1. $700 cm^3$

  2. $750 cm^3$

  3. $800 cm^3$

  4. $850 cm^3$


Correct Option: B
Explanation:

Given, perimeter $=4a=20$cm
$\therefore a=5$ cm
Area $=a^2=25  cm^2$
Volume $=$ Area $\times$ Height
Volume $=25 \times 30$
Volume $=750  cm^3$

The base of a right prism is an equilateral triangle of edge $12$m. If the volume of the prism is $288\sqrt 3m^3$, then its height is:

  1. $6$m

  2. $8$m

  3. $10$m

  4. $12$m


Correct Option: B
Explanation:

length of Equilateral triangle $= 12 m$
Area of equilateral triangle = $\displaystyle \frac{\sqrt{3}}{4}a^2$ = $\displaystyle \frac{\sqrt{3}}{4}(12)^2$ = $36\sqrt{3}$
Volume of prism = $288\sqrt{3} m^3$ = Area of triangle X height
$288\sqrt{3} m^3$ = $36 \sqrt{3} \times$ height
$\therefore $ Height $= 8 m$