Question

A sphérical báll of salt is dissolving in water in such a manner that the ráte of décrease of the volume at any iñstant is propôrtional to the surface. Prove that the radius is decréasing at a coñstant rate.​

Answer 1

Answer:

Here

P=

20,000

Rs.,

n=

3

years,

R=

8

Amount (A)=

P

(1+

100

R

)

n

=

20000

(1+

100

8

)

3

=

20000

(

25

27

)

3

=

25×25×25

20000×(27)

3

=

25

32×(27)

3

=

25,194.24

Rs.

Compound interest

(C.I)=

A−

P=

25,194.24−

20,000=

5,194.24

Hence, the compound interest is Rs. 5,

Answer 2

$\huge\mathfrak\red{Given::}$

$\text{A sphérical báll of salt is dissolving in water in} \\ \text{such a manner that the ráte of décrease of the} \\ \text{volume at any iñstant is propôrtional to the surface. } \\ \text{Prove that the radius is decréasing at a coñstant rate.}$

$\huge\mathfrak\red{Solution::}$

$\bf{ \frac{dv}{dt} } = k \times surface \: area \\ \\ \bf{v = \frac{4}{3}\pi \: {r}^{3} } \\ \\ \bf{sa = 4\pi {r}^{2} } \\ \\ \bf{ \frac{d( \frac{4}{3} \pi {r}^{3}) }{dt} } = k \times 4\pi {r}^{2} \\ \\ \bf{ \frac{4\pi \times d {r}^{3} }{3 \times dt} = 4\pi \: k {r}^{2} } \\ \\ \bf{ \frac{4\pi}{3} \times 3 {r}^{2} \times \frac{dr}{dt} = 4\pi \: k {r}^{2} } \\ \\ \bf{ \frac{dr}{dt} = k } \\ \\ \boxed{\mathfrak\green{Hence \: Proven}}$