Question

You make 2 quarts of salsa for a party. The recipe calls for 5 teaspoons of lime juice per quart, but you had accidentally put in 5 tablespoons per quart. You decide to feed your guests the salsa anyway. Assume that the guests take a quarter cup of salsa per minute and that you replace what was taken with chopped tomatoes and onions without any lime juice. [ 1 quart = 4 cups and \(1 \mathrm{~Tb}=3\) tsp.] a. Write the differential equation and initial condition for the amount of lime juice as a function of time in this mixture-type problem. b. Solve this initial value problem. c. How long will it take to get the salsa back to the recipe's suggested concentration?

Short answer

It will approximately take 18.42 minutes to get the salsa back to the recipe's suggested concentration.

Step-by-step solution

Establishing the differential equation and initial condition

Let \( y(t) \) be the amount of lime juice as a function of time. Since we replace 1/4 cup of salsa per minute, the rate out is \( -\frac{y(t)}{4} \frac{1}{4} \) or \( -\frac{y(t)}{16} \) cups per minute. There is no lime juice coming it. Thus, the differential equation can be written as: \( \frac{dy}{dt} = -\frac{y}{16} \). For the initial 2 quarts of salsa with 5 tablespoons of lime juice each, is a total of 10 tablespoons of lime juice. Therefore, our initial condition is \( y(0) = 10 \times 3 = 30 \) tsp. This is because 1 tablespoon is equal to 3 tsp.

Solve the initial value problem

The differential equation \( \frac{dy}{dt} = -\frac{y}{16} \) is a simple linear differential equation and can be solved using the integrating factor method. Doing this, we find the solution as \( y(t) = C \times e^{-\frac{t}{16}} \). Using the initial condition \( y(0) = 30 \) tsp., it can be found that \( C = 30 \). So, the solution to the initial value problem is \( y(t) = 30e^{-\frac{t}{16}} \).

Find the time it takes to reach the recipe's suggested lime juice concentration

The recipe suggests 5 tsp of lime juice per quart. So, for 2 quarts of salsa, there should be 10 tsp. of lime juice. This implies setting \( 30e^{-\frac{t}{16}} = 10 \) and solving for time \( t \). It can be found by taking natural logarithms of both sides and solving for \( t \) as \( t = -16 \ln(\frac{1}{3}) \) minutes, which is approximately 18.42 minutes.

Key concepts

Initial Value Problem

An Initial Value Problem (IVP) involves determining a function that satisfies a given differential equation and meets specific initial conditions.
Basically, you're figuring out how some quantity changes over time, starting from a known initial amount.
This is essential in many real-world applications, such as engineering and physics, where we often start a process at a known state.
In this exercise, the IVP requires us to find out how lime juice reduces in the salsa over time.
The salsa begins with a specific initial amount of 30 teaspoons of lime juice, due to the excess added by mistake.
Our task is to apply the differential equation, which represents the rate of change of lime juice content as some of it is being replaced, to find out how the lime juice level decreases over time.
The IVP here dictates two primary components:

• **Differential Equation**: It models the rate of change. In our case, it’s \[ \frac{dy}{dt} = -\frac{y}{16} \] where \( y(t) \) is the amount of lime juice over time.
• **Initial Condition**: This anchors the problem with a known starting value, \( y(0) = 30 \) teaspoons for our salsa case.

With both in place, solving the IVP provides us a function \( y(t) \) that models how lime juice amount changes in the salsa over time.

Exponential Decay

Exponential decay describes a process where a quantity reduces at a rate proportional to its current value.
This concept is common in situations where the decrease is rapid at first and slows over time.
It is the opposite of exponential growth (like compound interest) and is found naturally in phenomena such as radioactive decay.In our salsa problem, the excess lime juice undergoes exponential decay as the guests remove salsa and it’s replaced without lime juice.
The rate of change of lime juice can be expressed through the differential equation:
\[ \frac{dy}{dt} = -\frac{y}{16} \] Here, the negative sign indicates that the lime juice amount is decreasing over time.
Solving this gives a function of the form:
\[ y(t) = C \times e^{-\frac{t}{16}} \] The exponential term \( e^{-\frac{t}{16}} \) showcases how the lime juice decays over time at every instance.
As time passes, the exponent lowers the expression, indicating reduced lime juice content.
This type of decay ensures that the juice content decreases in a predictable first-order fashion, which is typical for natural decaying processes.

Integrating Factor Method

The Integrating Factor Method is a popular technique for solving linear first-order differential equations.
It's a straightforward method that transforms a differential equation into an easily integrable form.
By applying this method, you can easily solve for functions that are not directly integrable.For our lime juice problem, the differential equation is:
\[ \frac{dy}{dt} = -\frac{y}{16} \] To handle this using the Integrating Factor Method, the approach is:

• First, identify the integrating factor \( \mu(t) \) which is typically \( e^{\int P(t)\,dt} \).
• Here, since the equation is \( dy/dt + \frac{1}{16}y = 0 \), \( P(t) = \frac{1}{16} \), which leads to \( \mu(t) = e^{t/16} \).
• Multiply through by the integrating factor to make the left side of the equation a derivative of a product \( \frac{d}{dt}(y \cdot e^{t/16}) = 0 \).
• Integrate both sides to solve for \( y \).

Applying this method means the process unfolds as:
\[ y(t) = C \times e^{-\frac{t}{16}} \] This solution aligns with our expected form for exponential decay, providing a clear and methodical approach to solve the initial value problem.