Q67 / 150 · MHT CET 2026 (16 April Shift 1) · Mathematics
The rate of reduction is proportional to the square root of a persons existing assets at that moment. If his assets at the beginning are 10000 and they dwindle down to 5625 in 2 years, then the person will be bankrupt in
- A4 years
- B6 years
- Correct answerC8 years
- D10 years
Solution
Let $A$ be the assets at time $t$. The rate of reduction is proportional to the square root of the existing assets.
$\dfrac{dA}{dt} = -k\sqrt{A}$
Separating the variables and integrating:
$\int \dfrac{dA}{\sqrt{A}} = \int -k dt$
$2\sqrt{A} = -kt + C$
At $t = 0$, $A = 10000$
$2\sqrt{10000} = -k(0) + C \Rightarrow C = 200$
At $t = 2$ years, $A = 5625$
$2\sqrt{5625} = -k(2) + 200$
$2(75) = -2k + 200$
$150 = -2k + 200 \Rightarrow 2k = 50 \Rightarrow k = 25$
The equation for the assets at time $t$ becomes:
$2\sqrt{A} = -25t + 200$
For the person to be bankrupt, the assets $A$ must be $0$.
$2\sqrt{0} = -25t + 200$
$25t = 200 \Rightarrow t = 8$ years
Answer: 8 years
$\dfrac{dA}{dt} = -k\sqrt{A}$
Separating the variables and integrating:
$\int \dfrac{dA}{\sqrt{A}} = \int -k dt$
$2\sqrt{A} = -kt + C$
At $t = 0$, $A = 10000$
$2\sqrt{10000} = -k(0) + C \Rightarrow C = 200$
At $t = 2$ years, $A = 5625$
$2\sqrt{5625} = -k(2) + 200$
$2(75) = -2k + 200$
$150 = -2k + 200 \Rightarrow 2k = 50 \Rightarrow k = 25$
The equation for the assets at time $t$ becomes:
$2\sqrt{A} = -25t + 200$
For the person to be bankrupt, the assets $A$ must be $0$.
$2\sqrt{0} = -25t + 200$
$25t = 200 \Rightarrow t = 8$ years
Answer: 8 years
Hence, the correct answer is option: 'C'.