Applications of Differential Equations

If the population grows at the rate of $5\%$ per year, then the time taken for the population to become double is (Given $\log 2=0·6912$)

The bacteria increases at the rate proportional to the number of bacteria present. If the original number $N_0$ doubles in $4$ hours, then the number of bacteria in $12$ hours will be

The rate of decay of certain substance is directly proportional to the amount present at that instant. Initially, there are $27 \mathrm{gms}$ of certain substance and $3$ hours later it is found that $8 \mathrm{gms}$ are left, then the amount left after one more hour is

For next two question please follow the same

Consider a tank which initially holds $V_0$ liter of brine that contains $a$ lb of salt. Another brine solution, containing $b$ lb of salt per liter is poured into the tank at the rate of $e$ $\mathrm{L}/\min$ while, simultaneously, the well-stirred solution leaves the tank at the rate of $f$ $\mathrm{L}/\min$. The problem is to find the amount of salt in the tank at any time $t$.
Let $Q$ denote the amount of salt in the tank at any time. The time rate of change of $Q$, $\frac{dQ}{dt}$, equals the rate at which salt enters the tank at the rate of $be$ $\mathrm{lb}/\min$. To determine the rate at which salt leaves the tank, we first calculate the volume of brine in the tank at any time $t$, which is the initial volume $V_0$ plus the volume of brine added $et$ minus the volume of brine removed $ft$. Thus, the volume of brine at any time is
$V_0+et-ft ....\left(a\right)$
The concentration of salt in the tank at any time is $Q/\left(V_0+et-ft\right)$ from which it follows that salt leaves the tank at the rate of $f\left(\frac{Q}{V_0+et-ft}\right)\mathrm{lb}/\min$. Thus, 

$\frac{dQ}{dt}=be-f\left(\frac{Q}{V_0+et-ft}\right) .....\left(b\right)$

or $\frac{dQ}{dt}+\frac{f}{V_0+et-ft}Q=be$

 A $50 \mathrm{L}$ tank initially contains $10 \mathrm{L}$ of fresh water. At $t=0$, a brine solution containing $1 \mathrm{lb}$ of salt per gallon is poured into the tank at the rate of $4 \mathrm{L}/\min$ while the well-stirred mixture leaves the tank at the rate of $2 \mathrm{L}/\min$. Then the amount of time required for overflow to occur is

A tangent drawn to the curve $\mathrm{y}=\mathrm{f}\left(\mathrm{x}\right)$ at $\mathrm{P}\left(\mathrm{x}, \mathrm{y}\right)$ cuts the $\mathrm{x} \text{\&} \mathrm{y}$ axis at $\mathrm{A} \text{and} \mathrm{B}$ respectively. If $\mathrm{BP} : \mathrm{AP}=3 : 1$ and $\mathrm{f}\left(1\right)=1$, then the differential equation of curve is

In a bank, principal increases continuously at the rate of $6\%$ per year, the time required to double Rs.$6000$ is

Which equation represents the consumption function?

What represents the saving function in the short run?

What function would you use in a spreadsheet to determine the future value of an asset?

By what factor did the world population increase between 1950 and 2000?

What is the formula for calculating Gross Domestic Product (GDP) at market prices using the income method?

Macroeconomics emerged as a distinct field of study during which event?

Which term is used to describe a graph showing the relationship between the price of a commodity and the quantity demanded?

When is a consumer said to be in equilibrium?

What does the slope of the budget line represent in consumer theory?

Which of the following best describes the optimal choice of a consumer?

Which of the following equations represents a consumer's budget constraint?

Which technique involves comparing actual performance with predetermined standards?

Which analysis technique is used to determine the financial health of different business segments?

The temperature $T\left(t\right)$ of a body at time $t=0$ is ${160}^{°} F$ and it decreases continuously as per the differential equation $\frac{dT}{dt}=-K\left(T-80\right)$, where $K$ is positive constant. If $T\left(15\right)={120}^{°} F$, then $T\left(45\right)$ is equal to