Problem 28
Question
Choose three numbers \(x, y\), and \(z\) so that their sum is equal to 75 and their product is maximal.
Step-by-Step Solution
Verified Answer
The numbers are 25, 25, and 25.
1Step 1: Define Variables and Set Constraint
We need to find three numbers, denoted by \(x\), \(y\), and \(z\), such that their sum is 75. Thus, the constraint is given by the equation \(x + y + z = 75\).
2Step 2: Express One Variable in Terms of Others
To simplify, express \(z\) in terms of \(x\) and \(y\) using the constraint: \(z = 75 - x - y\).
3Step 3: Formulate the Objective Function
The objective is to maximize the product \(xyz\). Substitute \(z = 75 - x - y\) into this product to get \(f(x, y) = x \, y \, (75 - x - y)\). Simplifying, we have \(f(x, y) = 75xy - x^2y - xy^2\).
4Step 4: Use Symmetry to Simplify Further
To simplify the problem, assume \(x = y = z\). Since their sum is 75, we have \(3x = 75\) which gives \(x = 25\). Hence, \(y = 25\) and \(z = 25\).
5Step 5: Verify Maximal Product
Calculate the product for \(x = 25\), \(y = 25\), and \(z = 25\): \(25 \, \cdot \, 25 \, \cdot \, 25 = 15625\). Since any other distribution would make one number decrease (which decreases the product), this distribution is optimal.
Key Concepts
Maxima and MinimaConstraintsAlgebraic Manipulation
Maxima and Minima
In mathematics, optimization problems involve finding the maximum or minimum value of a function within certain constraints. When we deal with maxima and minima, we are interested in determining where these extreme values occur. For the provided exercise, this means we are trying to maximize the product of three numbers under given conditions.
To find the maximum product of three numbers whose sum equals 75, we must first define our objective function. The objective function here is the product of the numbers, represented as \(xyz\). By expressing one variable in terms of the others using the constraint (\(x + y + z = 75\)), we reduce the dimension and complexity of our task. This leads to the expression \(f(x, y) = x \, y \, (75 - x - y)\). By examining where this function reaches a peak, we find the maximum product value. In this problem, the task simplifies by assuming symmetry, meaning we can assume all numbers are equal, leading us to the conclusion without dealing with more complex calculations.
To find the maximum product of three numbers whose sum equals 75, we must first define our objective function. The objective function here is the product of the numbers, represented as \(xyz\). By expressing one variable in terms of the others using the constraint (\(x + y + z = 75\)), we reduce the dimension and complexity of our task. This leads to the expression \(f(x, y) = x \, y \, (75 - x - y)\). By examining where this function reaches a peak, we find the maximum product value. In this problem, the task simplifies by assuming symmetry, meaning we can assume all numbers are equal, leading us to the conclusion without dealing with more complex calculations.
Constraints
Constraints are essential in optimization problems because they define the boundary within which we must find a solution. For this problem, the main constraint is that the sum of the three numbers must equal 75. This constraint helps to simplify the problem by allowing us to express one variable in terms of the others.
By setting the constraint \(x + y + z = 75\), we are required to express \(z\) as \(z = 75 - x - y\). This step is crucial because it reduces the problem to two variables, making the objective function easier to work with. With the constraint in place, any choice of \(x\) and \(y\) automatically determines \(z\). This ensures that all cases considered are valid under the condition that their sum is always 75.
Constraints guide us in forming valid solutions, and understanding their role is key to solving optimization problems effectively.
By setting the constraint \(x + y + z = 75\), we are required to express \(z\) as \(z = 75 - x - y\). This step is crucial because it reduces the problem to two variables, making the objective function easier to work with. With the constraint in place, any choice of \(x\) and \(y\) automatically determines \(z\). This ensures that all cases considered are valid under the condition that their sum is always 75.
Constraints guide us in forming valid solutions, and understanding their role is key to solving optimization problems effectively.
Algebraic Manipulation
Algebraic manipulation is the process of rearranging and simplifying algebraic expressions and equations to make them easier to work with. In our optimization problem, algebraic manipulation is used to create the objective function and solve for the variables that maximize it.
Initially, we used the constraint \(z = 75 - x - y\) to substitute the variable \(z\) in the product expression \(xyz\). This resulted in the expression \(f(x, y) = x \, y \, (75 - x - y)\). Expanding this formula through algebraic manipulation gives \(75xy - x^2y - xy^2\).
Further simplification involves assuming that the three numbers are equal if possible. This assumption opens the way to easily determine the maximum value by setting \(x = y = z\) and solving for one variable, leading to even simpler calculations. Algebraic manipulation in this context leverages symmetry and logical assumptions to make the problem manageable and the solution clearer.
Initially, we used the constraint \(z = 75 - x - y\) to substitute the variable \(z\) in the product expression \(xyz\). This resulted in the expression \(f(x, y) = x \, y \, (75 - x - y)\). Expanding this formula through algebraic manipulation gives \(75xy - x^2y - xy^2\).
Further simplification involves assuming that the three numbers are equal if possible. This assumption opens the way to easily determine the maximum value by setting \(x = y = z\) and solving for one variable, leading to even simpler calculations. Algebraic manipulation in this context leverages symmetry and logical assumptions to make the problem manageable and the solution clearer.
Other exercises in this chapter
Problem 28
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