Water flows from the base of a rectangular ta | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(B) Let $x$ be the depth of water at time $t$. Given that the rate of flow is proportional to the square root of the depth,we have $\frac{dx}{dt} = -k

Vedclass · Accepted Answer

$0.25$

Vedclass · Answer

(B) Let $x$ be the depth of water at time $t$. Given that the rate of flow is proportional to the square root of the depth,we have $\frac{dx}{dt} = -k\sqrt{x}$,where $k > 0$ is a constant (negative sign indicates decreasing depth).Separating the variables,we get $\frac{dx}{\sqrt{x}} = -k dt$.Integrating both sides,we get $\int x^{-1/2} dx = \int -k dt$,which gives $2\sqrt{x} = -kt + C$.At $t = 0$,the initial depth $x = 16 \ m$. Substituting these values,$2\sqrt{16} = -k(0) + C \Rightarrow C = 8$.So,the equation becomes $2\sqrt{x} = -kt + 8$.At $t = 2 \ hours$,the depth $x = 4 \ m$. Substituting these values,$2\sqrt{4} = -k(2) + 8 \Rightarrow 4 = -2k + 8 \Rightarrow 2k = 4 \Rightarrow k = 2$.Thus,the equation is $2\sqrt{x} = -2t + 8$,or $\sqrt{x} = 4 - t$.For $t = 3.5 \ hours$,$\sqrt{x} = 4 - 3.5 = 0.5$.Squaring both sides,$x = (0.5)^2 = 0.25 \ m$.

Water flows from the base of a rectangular tank of depth $16 \ m$. The rate of flow of the water is proportional to the square root of the depth at any time $t$. If the depth is $4 \ m$ when $t = 2 \ hours$,then after $3.5 \ hours$,the depth (in meters) is:

Similar Questions

The population of a town increases at a rate proportional to the population at that time. If the population increases from $40,000$ to $80,000$ in $40$ years,then the population in another $40$ years will be (in $,000$)

In a culture,the bacteria count is $1,00,000$. The number increases by $10 \%$ in $2$ hours. In how many hours will the count reach $2,00,000$,if the rate of growth of bacteria is proportional to the number present?

The decay rate of radioactive material at any time $t$ is proportional to its mass at that time. The mass is $27 \text{ grams}$ when $t=0$. After $3 \text{ hours}$,it was found that $8 \text{ grams}$ are left. Then the substance left after one more hour is

Water at $100^{\circ} C$ cools in $10 \text{ minutes}$ to $80^{\circ} C$ in a room temperature of $25^{\circ} C$. What will be the temperature of water after $20 \text{ minutes}$ (in $^{\circ} C$)?

$A$ spherical balloon expands at a rate proportional to its surface area. Initially, its radius is $2 \text{ cm}$ and $5$ minutes later it increases to $7 \text{ cm}$. What will be the surface area of the spherical balloon after $12$ minutes (in $\text{ cm}^2$)?

Vedclass Test Series

Exam Paper Generator

Online Exam Module