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Chapter 12: Problem 41

Given the product of two numbers and their ratio, to find the roots: Let \(A, E\), be the two roots, \(A E=B, A: E=\) \(S: R\). Show that \(R: S=B: A^{2}\) and \(S: R=B: E^{2}\). Vi猫te's example has \(B=20, R=1, S=5\). Show in this case that \(A=10\) and \(E=2\). (Jordanus has the same problem but with different numbers.)

Short Answer

Expert verified
Answer: When B=20, R=1, and S=5, the values of A and E are 10 and 2, respectively.

Step by step solution

01

Write the given relations

We are given the following relations: 1. \(A E = B\) 2. \(\frac{A}{E} = \frac{S}{R}\)
02

Find the relationship between \(R: S\) and \(B: A^{2}\), and \(S: R\) and \(B: E^{2}\)

From the second relation, we can express \(A\) and \(E\) in terms of \(R\) and \(S\): \(A = \frac{S * E}{R}\) Now, substitute the value of A obtained from the second relation into the first relation: \(E * \frac{S * E}{R} = B\) Rearrange to find the relationship between \(R:S\) and \(B:A^2\): \(\frac{R}{S} = \frac{B}{E^2}\) Similarly, for the relationship between \(S:R\) and \(B:E^2\), \(E = \frac{R * A}{S}\) Now, substitute the value of E obtained from the second relation into the first relation: \(A * \frac{R * A}{S} = B\) Rearrange to find the relationship between \(S:R\) and \(B:A^2\): \(\frac{S}{R} = \frac{B}{A^2}\)
03

Use the example given to find the values of \(A\) and \(E\)

We are given \(B = 20\), \(R = 1\), and \(S = 5\). We use these values to find \(A\) and \(E\) using the relationships obtained above. \(\frac{S}{R} = \frac{B}{A^2}\) \(\frac{5}{1} = \frac{20}{A^2}\) => \(A^2 = 4\) => \(A = 10\) Now, use the second relationship to find the value of \(E\): \(\frac{R}{S} = \frac{B}{E^2}\) \(\frac{1}{5} = \frac{20}{E^2}\) => \(E^2 = 100\) => \(E = 2\) Therefore, the values of \(A\) and \(E\) are 10 and 2, respectively.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Algebraic Equations
Algebraic equations are foundational to understanding mathematical relationships and they play a crucial role in problem solving across various fields. In essence, an algebraic equation is a statement of equality between two expressions involving constants and variables. For instance, the equation used in our problem, \( A \times E = B \) expresses the product of two unknown numbers 鈥� A and E 鈥� equating to a known number B.

When tackling problems involving algebraic equations, one effective approach is to manipulate these equations to express one variable in terms of others, which then allows for substitution into other equations to find a solution. In our example, we rearranged the given equation to isolate the variable E, leading to a new expression that could be used in further calculations. Such skills in manipulating algebraic equations are not only essential for solving textbook problems but also for critical thinking and analysis in real-world applications.
Ratios and Proportions
Ratios and proportions are important concepts in mathematics and are especially useful when comparing quantities. A ratio is a comparison between two numbers showing how many times the first number contains the second. In our exercise, we look at the ratio \( A:E = S:R \) which compares two unknowns, A and E, in relation to two knowns, S and R. Proportions, on the other hand, state that two ratios are equivalent.

Understanding how to work with ratios and proportions allows for the solving of problems where direct measurement is not possible, or when working with different units of measurement. In our problem-solving process, we used proportional relationships to express one variable in terms of another. This is an example of how the concepts of ratios and proportions can simplify complex problems by setting up relations that can be manipulated algebraically to extract useful information.
Mathematical Roots
Mathematical roots are solutions to the equation where a number or expression is raised to a power and set equal to another number. Basically, finding a root is the inverse operation of exponentiation. In our exercise, we had to calculate the square roots of numbers as part of finding the solution. The square root, which is the most common type of root, answers the question: 'what number, when multiplied by itself, will give me the original number?'

For instance, with \( A^2 = 4 \) from the problem, we consider what number squared equals 4, and we rightly find that \( A = 10 \) (note that we consider the positive root in the context of this problem). Understanding roots is essential for solving quadratic equations and forms the basis for more advanced mathematics, including calculus. Recognizing and calculating roots is not only crucial for academic purposes but also for practical applications, such as in engineering and sciences, where they are used to model and solve real-life problems.

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Most popular questions from this chapter

Use Ferrari's method to solve the quartic equation \(x^{4}+\) \(4 x+8=10 x^{2}\). Begin by rewriting this as \(x^{4}=10 x^{2}-\) \(4 x-8\) and adding \(-2 b x+b^{2}\) to both sides. Determine the cubic equation that \(b\) must satisfy so that each side of the resulting equation is a perfect square. For each solution of that cubic, find all solutions for \(x\). How many different solutions to the original equation are there?

This problem is from the Treviso Arithmetic, the first printed arithmetic text, dated 1478: The Holy Father sent a courier from Rome to Venice, commanding him that he should reach Venice in 7 days. And the most illustrious Signoria of Venice also sent another courier to Rome, who should reach Rome in 9 days. And from Rome to Venice is 250 miles. It happened that by order of these lords the couriers started their journeys at the same time. It is required to find in how many days they will meet, and how many miles each will have traveled \({ }^{37}\)

There is a strange journey appointed to a man. The first day he must go \(1 \frac{1}{2}\) miles, and every day after the first he must increase his journey by \(\frac{1}{6}\) of a mile, so that his journey shall proceed by an arithmetical progression. And he has to travel for his whole journey 2955 miles. In what number of days will he end his journey?

Use Cardano's formula to solve \(x^{3}=6 x+6\)

A fountain has two basins, one above and one below, each of which has three outlets. The first outlet of the top basin. fills the lower basin in two hours, the second in three hours, and the third in four hours. When all three upper outlets are shut, the first outlet of the lower basin empties it in three hours, the second in four hours, and the third in five hours. If all the outlets are opened, how long will it take for the lower basin to fill?
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