$m, n$, and $p$ are differentiable with respect to time and are related by the equation $m^2+mn=p$. 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

04-11-2026
Mathematics

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$m, n$, and $p$ are differentiable with respect to time and are related by the equation $m^2+mn=p$. 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 · 2026-11-04T22:59:19-05:00

Identify the value of $n$ at the given instant: $m=3$ and $p=21$ leads to $n=4$.
Differentiate the relationship $m^2 + mn = p$ with respect to time $t$ using the product rule.
Substitute the known values $m=3, n=4, \frac{dm}{dt}=2, \frac{dn}{dt}=-1$ into the derivative equation.
Calculate the final rate of change for $p$, obtaining $\boxed{17}$ units per minute.

Explanation

Problem Analysis and Determining Instantaneous State

The variables $m$, $n$, and $p$ are related by the equation $m^2 + mn = p$. We are given that $m$ is increasing at a constant rate of $\frac{dm}{dt} = 2$ units/min and $n$ is decreasing at a constant rate of $\frac{dn}{dt} = -1$ unit/min. We need to find the rate of change of $p$, $\frac{dp}{dt}$, at the moment when $m=3$ and $p=21$. First, we must determine the value of $n$ at this instant by substituting the known values into the given equation:
$3^2 + 3n = 21$
$9 + 3n = 21$
$3n = 12$
$n = 4$
Thus, at the specific instant, $m=3$, $n=4$, and $p=21$.

Differentiating with Respect to Time

To find the rate of change of $p$, we differentiate the given relationship $m^2 + mn = p$ with respect to time $t$. Using the power rule for $m^2$, and the product rule for the term $mn$, we obtain:
$\frac{d}{dt}(m^2 + mn) = \frac{d}{dt}(p)$
$2m \frac{dm}{dt} + \left( m \frac{dn}{dt} + n \frac{dm}{dt} \right) = \frac{dp}{dt}$
This expression links the rates of change of each variable involved in the equation.

Calculating the Rate of Change of p

Now we substitute the known values $m=3$, $n=4$, $\frac{dm}{dt}=2$, and $\frac{dn}{dt}=-1$ into the differentiated equation:
$\frac{dp}{dt} = 2(3)(2) + (3(-1) + 4(2))$
$\frac{dp}{dt} = 12 + (-3 + 8)$
$\frac{dp}{dt} = 12 + 5$
$\frac{dp}{dt} = 17$
The rate of change of $p$ at this instant is 17 units per minute.

Final Answer

Based on the calculations above, the rate of change of $p$ when $m=3$ and $p=21$ is $\boxed{17}$ units per minute.

Examples

In engineering, this type of calculation is vital when variables are interdependent, such as in a control system where a pressure $p$ depends on flow rate $m$ and viscosity $n$. If you know how the inputs change over time, you can instantly determine how the system pressure will evolve. This ensures that safety thresholds are not exceeded as equipment operates.