A solid metallic right circular cone 20cm high and whose vertical… - Vidyadip

Question

A solid metallic right circular cone 20cm high and whose vertical angle is 60°, is cut into two parts at the middle of its height by a plane parallel to its base. If the frustum so obtained be drawn into a wire of diameter $\frac{1}{12}\text{cm},$ then find the length of the wire.

✓

Answer

We have,
Height of the solid metallic cone, H = 20cm,
Height of the frustom, $\text{h}=\frac{20}{2}=10\text{cm}$ and
Radius of the wire $=\frac{1}{24}\text{cm}$
Let the length of the wire be l, BG = r and BO = R.
In $\triangle\text{AEG},$
$\tan30^\circ=\frac{\text{EG}}{\text{AG}}$
$\Rightarrow\frac{1}{\sqrt3}=\frac{\text{r}}{\text{H}-\text{h}}$
$\Rightarrow\frac{1}{\sqrt3}=\frac{\text{r}}{\text{20}-\text{10}}$
$\Rightarrow\text{r}=\frac{10}{\sqrt3}\text{cm}$
Also, in $\triangle\text{ABD},$
$\tan30^\circ=\frac{\text{BD}}{\text{AD}}$
$\Rightarrow\frac{1}{\sqrt3}=\frac{\text{R}}{\text{H}}$
$\Rightarrow\frac{1}{\sqrt3}=\frac{\text{R}}{\text{20}}$
$\Rightarrow\text{R}=\frac{20}{\sqrt3}\text{cm}$
Now,
Volume of the wire = Volume of the frustum
$\Rightarrow\pi\Big(\frac{1}{24}\Big)^2\text{l}=\frac{1}{3}\pi\text{h}\big(\text{R}^2+\text{r}^2+\text{Rr}\big)$
$\Rightarrow\frac{\text{l}}{576}=\frac{1}{3}\times10\times\bigg[\Big(\frac{20}{\sqrt3}\Big)^2+\Big(\frac{10}{\sqrt3}\Big)^2+\Big(\frac{20}{\sqrt3}\Big)\Big(\frac{10}{\sqrt3}\Big)\bigg]$
$\Rightarrow\text{l}=\frac{576}{3}\times10\times\Big[\frac{400}{3}+\frac{100}{3}+\frac{200}{3}\Big]$
$\Rightarrow\text{l}=\frac{576}{3}\times10\times\frac{700}{3}$
$\Rightarrow\text{l}=448000\text{cm}$
$\therefore\text{l}=4480\text{m}$
So, the length or the wire Is 4460m.

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "5 Marks Questions" from Volume and Surface Areas of Solids→Volume and Surface Areas of Solids — all question types→Maths STD 10 question papers→CBSE Board STD 10 question papers→CBSE Board question paper generator→

Similar questions

Find the area and perimeter of an isosceles right-angled triangle, each of whose equal sides measures 10cm. $\big[\text{Given}:\sqrt{2}=1.41\big]$

If three times the larger of two numbers is divided by the smaller, we get 4 as the quotient and 8 as the remainder. If five times the smaller is divided by the larger, we get 3 as the quotient and 5 as the remainder. Find the numbers.

Prove the following identities:
$\frac{\big(1+\tan^2\theta\big)\cot\theta}{\text{cosec}^2\theta}=\tan\theta$

Solve the following quadratic equations by factorization:
$\frac{\text{x}-3}{\text{x}+3}-\frac{\text{x}+3}{\text{x}-3}=\frac{48}{7},$ $\text{x}\neq3,\text{x}\neq-3$

The mean of the following frequency distribution is 57.6 and the sum of the observations is 50.

Class	0-20	20-40	40-60	60-80	80-100	100-120
Frequency	7	$f_1$	12	$f_2$	8	5

??The following table gives the life-time (in days) of 100 electric bulbs of a certain brand.

Life-time (in days)	Less than 50	Less than 100	Less than 150	Less than 200	Less than 250	Less than 300
Number of bulbs	7	21	52	79	91	100

From this table, construct the frequency distribution table.

Solve the following systems of equations:
$\text{x}+\text{y}=2\text{xy},$
$\frac{\text{x}-\text{y}}{\text{xy}}=6,\text{x}\neq0,\text{y}\neq0.$

Find the perimeter of the s ded region in the figure, if ABCD is a square of side 14cm and APB and CPD are semicircles.

Find the mode of the following distribution:

Class interval	10-14	14-18	18-22	22-26	26-30	30-34	34-38	38-42
Frequency	8	6	11	20	25	22	10	4

Which term of the AP:
-2, -7, -12, ........ will be -77? Find the sum of this AP upto the term -77.