Question

If the volumes of two cones are in the ratio of 1:4 and ther diameters are in the ratio of 4:5, then the ratio of their heights are
(a) 25 : 64
(b) 3:5
(c) 5:8
(d) none of these​

Answer 1

Answer:

(a) 25:64 is a right answer

Answer 2

Answer:

$\bigstar{\bold{Option\:a:\:25:64}}$

Step-by-step explanation:

$\Large{\underline{\underline{\bf{Given:}}}}$

• Volume of two cones are in the ratio 1 : 4
• Diameters are in the ratio 4 : 5

$\Large{\underline{\underline{\bf{To\:Find:}}}}$

• Ratio of the heights

$\Large{\underline{\underline{\bf{Solution:}}}}$

→ Given diameters are in the ratio 4 : 5

  Radius R/r = D/2 /d/2 = D/d = 4/5

→ Hence radii is in the ratio 4:5

  Let R be 4x

  Ler r be 5x

→ We know that the volume of the cone is given by the equation,

   Volume of a cone = 1/3 × π × r² × h

→ Hence by given datas,

  $\dfrac{Volume\:of\:first\:cone}{Volume\:of\:second\:cone}=\dfrac{\dfrac{1}{3}\pi R^{2}H }{\dfrac{1}{3} \pi r^{2}h }=\dfrac{1}{4}$

→ Cancelling 1/3 and π on both numerator and denominator

   $\dfrac{R^{2}H }{r^{2} h}=\dfrac{1}{4}$

→ Substitute value of R and r

  $\dfrac{(4x)^{2}H }{(5x)^{2}h } =\dfrac{1}{4}$

 

   $\dfrac{16x^{2}H }{25x^{2} h} =\dfrac{1}{4}$

→ Cancelling x² on both numerator and denominator

   $\dfrac{16H}{25h} =\dfrac{1}{4}$

   $\dfrac{H}{h} =\dfrac{1}{4}\times \dfrac{25}{16}$

  $\dfrac{H}{h} =\dfrac{25}{64}$

→ Hence ratio of heights is 25:64

  $\boxed{\bold{Ratio\:of\:heights=25:64}}$

→ Hence option a is correct

$\Large{\underline{\underline{\bf{Notes:}}}}$

→ Volume of a cone is given by the formula,

   Volume of cone = 1/3 × π × r² × h