Question

An increase in world production of processed fruit has led to an increase in fruit waste. One way of reducing this waste is to find useful waste byproducts. For example, waste from pineapples is reduced by extracting pectin from pineapple peels (pectin is commonly used as a thickening agent in jam and jellies, and it is also widely used in the pharmaceutical industry). Pectin extraction involves heating and drying the peels, then grinding the peels into a fine powder. The powder is next placed in a solution with a particular pH level \(H,\) for \(1.5 \leq H \leq 2.5,\) and heated to a temperature \(T\) (in degrees Celsius), for \(70 \leq T \leq 90 .\) The percentage of the powder \(F(H, T)\) that becomes extracted pectin is $$ \begin{aligned} F(H, T)=&-0.042 T^{2}-0.213 T H-11.219 H^{2}+7.327 T \\ &+58.729 H-342.684 \end{aligned} $$ a. It can be shown that \(F\) attains its absolute maximum in the interior of the domain \(D=\\{(H, T): 1.5 \leq H \leq 2.5\) \(70 \leq T \leq 90\\} .\) Find the \(\mathrm{pH}\) level \(H\) and temperature \(T\) that together maximize the amount of pectin extracted from the powder. b. What is the maximum percentage of pectin that can be extracted from the powder? Round your answer to the nearest whole number. (Source: Carpathian Journal of Food Science and Technology, Dec 2014 )

Short answer

Answer: The approximate pH level and temperature to maximize pectin extraction are 2.151 and 84.619°C, respectively. The maximum percentage of pectin extracted is approximately 15%.

Step-by-step solution

Calculate partial derivatives

In order to find the absolute maximum of \(F(H,T)\), we must first find the critical points of the function by calculating the partial derivatives with respect to \(H\) and \(T\). We first calculate the partial derivative with respect to \(H\): $$\frac{\partial F}{\partial H} = -0.213 T - 22.438 H + 58.729$$ Next, we calculate the partial derivative with respect to \(T\): $$\frac{\partial F}{\partial T} = -0.084 T^2 -0.213 H + 7.327$$

Find critical points

Now we will use these partial derivatives to find the critical points of the function by solving the following system of equations: $$ \begin{cases} \frac{\partial F}{\partial H} = 0 \\ \frac{\partial F}{\partial T} = 0 \end{cases} $$ Solving this system of equations, we get: $$H \approx 2.151, \quad T \approx 84.619$$ Therefore, to maximize the amount of pectin extracted from the fruit powder, the pH level should be approximately \(2.151\) and the temperature \(84.619\) degrees Celsius. #b. Finding the maximum percentage of pectin extracted#

Find the value of F(H, T) at the critical point

We have already found the critical point \((H, T) \approx (2.151, 84.619)\). Now, we need to substitute these values into the given function \(F(H, T)\) to find the maximum percentage of pectin extracted: $$ \begin{aligned} F(2.151, 84.619) \approx& -0.042 (84.619)^2 - 0.213 (2.151)(84.619) - 11.219 (2.151)^2 \\ &+ 7.327 (84.619) + 58.729 (2.151) - 342.684 \end{aligned} $$ Evaluating this expression, we get \(F(2.151, 84.619) \approx 15.005\).

Round the percentage

Rounding the obtained value to the nearest whole number, we find that the maximum percentage of pectin that can be extracted from the fruit powder is approximately \(15\%\).

Key concepts

Partial Derivatives

When analyzing functions of multiple variables, like the percentage of extracted pectin from pineapple peels given as a function of temperature and pH level, it's important to understand how the function's value changes as each individual variable changes. This is where partial derivatives come into play.

Imagine you are holding one variable constant and examining how the function changes with respect to another variable. In calculus, this is done by taking the partial derivative of the function with respect to that variable. For instance, if you have a function like \( F(H, T) \), then the partial derivative with respect to \( H \), denoted as \( \frac{\partial F}{\partial H} \), reflects the rate of change of \( F \) as \( H \) changes while keeping \( T \) constant.

Similarly, \( \frac{\partial F}{\partial T} \) represents the rate at which the function's value changes as the temperature \( T \) varies, with the pH level \( H \) held constant. In the given exercise, these partial derivatives were calculated to find the point where both derivatives are zero, which can potentially be the spot for a function's extreme value - its maximum or minimum.

Critical Points

Once the partial derivatives are in hand, the next step in optimization problems is to locate the critical points. These are the points where the partial derivatives equal zero, or where they do not exist. In real-world problems, such as maximizing pectin extraction, these critical points are often where you'll find the optimal conditions of the variables involved.

At a critical point, the function doesn't increase or decrease as you make a tiny nudge in any direction of the variables. Think of standing at the top of a hill (a maximum) or at the bottom of a valley (a minimum). Wherever you step, initially leads you down or up, respectively. To find these all-important points, you set each partial derivative to zero and solve the resulting equations simultaneously.

In the problem presented, critical points were found where both partial derivatives equal zero, which implies that no infinitesimal change in pH level or temperature would initially increase the pectin extraction yield. Solving this gives you specific values for \( H \) and \( T \), identifying the pH level and temperature at which the percentage yield of pectin is optimal.

Absolute Maximum of a Function

The absolute maximum of a function, as the name suggests, is the greatest value that the function reaches over its domain. It's a peak beyond which the function never ascends, an apex of optimization problems. In a multivariable context, such as the one we encounter with the function \( F(H, T) \), it represents the highest point over the entire landscape defined by its different variables.

To determine this maximum, after finding the critical points, we can either examine a plot of the function or calculate its value at critical points and compare it with the function's values at the boundaries of the domain. The highest of these values is the absolute maximum. It is important to note that not all functions have an absolute maximum, but in the case of pectin extraction, the problem has assured us that an absolute maximum exists within the given domain.

In the given exercise, after finding the critical point, the value of the function \( F(H, T) \) at that point was calculated, approximating the absolute maximum of pectin percentage that can be extracted. The result was then rounded to provide a practical answer to the real-world scenario, telling us the best-case scenario for pectin yield from the pineapple peels.