Problem 54 - Finite Mathematics and Applied Calculus

Question

There should be no more of ingredient \(\mathrm{A}\) (by weight) than ingredient \(\mathrm{B}\). (A) \(x-y=0\) (B) \(x-y \leq 0\) (C) \(x-y \geq 0\) (D) \(x+y \geq y\)

Step-by-Step Solution

Answer

The correct inequality to represent the condition "there should be no more of ingredient A (by weight) than ingredient B" is \(x-y \leq 0\). Thus, the answer is (B).

Step 1: Understand the problem

We are given that the weight of ingredient A is represented by "x" and the weight of the ingredient B is represented by "y". We need to find the inequality that says "no more of A than B".

Step 2: Check each option

We will now test each inequality to see if it matches the condition given in the problem. (A) \(x-y=0\) This means x is equal to y. In other words, both ingredients have the same weight. This is not our desired condition, as it doesn't allow for cases when A is less than B. (B) \(x-y \leq 0\) This inequality states that x is less than or equal to y. In other words, the weight of A is less than or equal to the weight of B, which is our desired condition. (C) \(x-y \geq 0\) This inequality states that x is greater than or equal to y. In other words, the weight of A is more or equal to the weight of B. This does not match our desired condition. (D) \(x+y \geq y\) Simplifying the inequality gives: \(x \geq 0\) This inequality states that x is greater than or equal to 0. In other words, the weight of A is non-negative. This inequality doesn't provide any information about B, so it doesn't match our desired condition.

Step 3: Identify the correct inequality

Based on our analysis, the correct inequality to represent the condition given in the problem is: \(x-y \leq 0\) Therefore, the answer is (B).

Key Concepts

Linear Inequalities

Understanding linear inequalities is essential for solving a wide range of mathematical problems. A linear inequality looks similar to a linear equation but uses inequality symbols like <, >, ≤, and ≥ instead of an equals sign. It represents a range of possible solutions, rather than just one specific value.

Let's delve into a typical example involving ingredients, where we see an inequality like
(B) \(x-y \leq 0\)
In this case, \(x\) and \(y\) signify the weight of two different ingredients, A and B, respectively. The inequality \(x-y \leq 0\) can be further interpreted as 'Ingredient A should weigh no more than Ingredient B.' If we were to graph this inequality on a coordinate plane, it would include all the points on the line \(y=x\) and all the points below it, since those points correspond to all cases where A weighs less than or equal to B.

Understanding the relationship between the variables and the inequality sign is crucial. If the inequality were flipped, \(x-y \geq 0\), it would imply that A should weigh the same as or more than B, which doesn't align with the given condition in the problem.

Finite Mathematics

Finite mathematics is a field that encompasses various topics like set theory, counting, probability, statistics, finance, and matrices. The key characteristic that defines finite mathematics is that it deals with discrete and limited sets of elements, contrasting with infinite or continuous entities found in calculus.

Linear inequalities, such as the one found in the exercise \(x-y \leq 0\), play a vital role within finite mathematics. They often appear in optimization and linear programming problems, which are common in business, economics, and management sciences. In these applications, inequalities are used to express constraints – limitations or requirements that must be met. For instance, in a recipe, you might have a constraint that limits the amount of one ingredient relative to another, as we saw with ingredients A and B. Thus, grasping inequalities is not only important for pure mathematical theory but also for practical applications in various fields.

Applied Calculus

Applied calculus is the use of calculus to solve real-world problems and is often encountered in fields such as physics, engineering, economics, and even biology. It involves techniques like differentiation and integration to model and analyze dynamic systems. While our ingredient problem does not require calculus as such, understanding the rates of change could be vital when scaling recipes or mixing substances.

In more advanced scenarios, if one was dealing with rates at which ingredients need to be mixed over time, or if there were more complex constraints involving changes over time – like temperature changes affecting ingredient compositions – applied calculus could certainly come into play. Instead of a static inequality, one might deal with differential inequalities, which involve functions, their derivatives, and inequality relations. The ability to understand and manipulate inequalities within the context of calculus is therefore a crucial skill, furthering one’s capability to tackle a broad range of problems.