$m, n$, and $p$ are differentiable with respect to time and are related by the equation $m^2+m n=p$. If $m$ is increasing at a constant rate of 2 units per minute and $n$ is decreasing at a constant rate of 1 unit per minute, at what rate is $p$ changing when $m=3$ and $p=21$?

Question

Kolpyu Kolpyu

04-11-2026
Mathematics

Answered

$m, n$, and $p$ are differentiable with respect to time and are related by the equation $m^2+m n=p$. If $m$ is increasing at a constant rate of 2 units per minute and $n$ is decreasing at a constant rate of 1 unit per minute, at what rate is $p$ changing when $m=3$ and $p=21$?

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GinnyAnswer GinnyAnswer · Answer 1 · 2026-11-04T23:22:34-05:00

First, determine the value of $n$ using the given equation $m^2 + mn = p$ with the values $m=3$ and $p=21$, which yields $n=4$.
Differentiate the relationship $m^2 + mn = p$ with respect to time $t$ using the product rule to obtain $\frac{dp}{dt} = (2m + n)\frac{dm}{dt} + m\frac{dn}{dt}$.
Substitute the known values $m=3, n=4, \frac{dm}{dt}=2$, and $\frac{dn}{dt}=-1$ into the differentiated equation.
Calculate the final rate of change, arriving at $\boxed{17}$ units/min.

Explanation

Analysis of Variables and Given Rates

The problem provides a relationship between three differentiable variables $m(t)$, $n(t)$, and $p(t)$ defined by the equation $m^2 + mn = p$. We are given that $m$ is increasing at a rate of $\frac{dm}{dt} = 2$ units/min and $n$ is decreasing at a rate of $\frac{dn}{dt} = -1$ unit/min. We need to find the rate of change of $p$, denoted as $\frac{dp}{dt}$, at the specific moment when $m = 3$ and $p = 21$.

Finding the Instantaneous Value of n

Before finding the rate of change of $p$, we must determine the value of $n$ at the instant when $m = 3$ and $p = 21$. Using the given relationship $m^2 + mn = p$, we substitute the known values:
$3^2 + 3n = 21$
$9 + 3n = 21$
Subtracting 9 from both sides:
$3n = 12$
Dividing by 3:
$n = 4$
Thus, at the given instant, $n = 4$.

Differentiating the Governing Equation

To find the rate of change $\frac{dp}{dt}$, we differentiate the relationship $m^2 + mn = p$ with respect to time $t$. Applying the power rule for $m^2$ and the product rule for $mn$, we get:
$\frac{d}{dt}(m^2) + \frac{d}{dt}(mn) = \frac{dp}{dt}$
$2m\frac{dm}{dt} + \left( \frac{dm}{dt}n + m\frac{dn}{dt} \right) = \frac{dp}{dt}$
Rearranging gives:
$\frac{dp}{dt} = (2m + n)\frac{dm}{dt} + m\frac{dn}{dt}$

Calculating the Rate of Change of p

Now, we substitute the known values into the differentiated expression:
$m = 3, \quad n = 4, \quad \frac{dm}{dt} = 2, \quad \frac{dn}{dt} = -1$
$\frac{dp}{dt} = (2(3) + 4)(2) + (3)(-1)$
$\frac{dp}{dt} = (6 + 4)(2) - 3$
$\frac{dp}{dt} = (10)(2) - 3$
$\frac{dp}{dt} = 20 - 3$
$\frac{dp}{dt} = 17$
Thus, the rate of change of $p$ at that moment is 17 units/min.

Final Answer

The rate of change of $p$ with respect to time when $m=3$ and $p=21$ is $\boxed{17}$ units/min.

Examples

In engineering, imagine controlling the pressure $p$ within a pressurized tank, where $p$ is influenced by two internal variables $m$ and $n$ representing the volumes of different gas components. Knowing how these components change over time allows an engineer to determine the rate of pressure fluctuation, ensuring the system operates safely without exceeding structural stress limits. This application of related rates is fundamental in fluid dynamics and mechanical systems design.