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Chapter 12: Problem 51

Two numbers and their geometric mean are given. Find the value of \(a\). 4 and \(a ; 14\)

Short Answer

Expert verified
The value of \(a\) is 49.

Step by step solution

01

Understand the problem

We need to use the concept of geometric mean. The geometric mean of two positive numbers \(x\) and \(y\) is \(g = \sqrt{xy}\), where \(g\) represents the geometric mean of these two numbers.
02

Create equation using given values

It is given that the geometric mean of 4 and \(a\) is 14. We set this up as an equation: 14 = \(\sqrt{4*a}\).
03

Solve the equation to find value of a

Squaring both sides of the equation, we get: 196 = 4*a. Dividing both sides by 4, we get: a = 196 / 4 = 49.

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Key Concepts

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

Algebra
Algebra is like the building blocks of mathematics. It lets us take numbers and use letters to symbolize them, creating formulas called expressions and equations. When you see letters like \(a\) or \(x\), they represent unknown values or variables that we need to find. With algebra, we can manipulate these expressions by following certain rules to simplify, rearrange, or solve them. This process involves understanding and applying different operations such as addition, subtraction, multiplication, and division to find the solution. In the context of this exercise, we used algebra to set up an equation that expresses the relationship between the numbers and their geometric mean.
Equation Solving
Whenever we encounter an equation, we are faced with a mathematical sentence that contains an unknown value we need to find. Equation solving is all about unraveling this mystery to find what those unknowns truly are. The method usually involves isolating the variable - which means getting the unknown value by itself on one side of the equation. This is typically done by performing various operations on both sides of the equation to keep the equality balanced. Here are a few tips to remember:
  • Always perform the same operation on both sides of the equation.
  • Use the inverse of operations to isolate the variable, for example, subtract if the equation adds, divide if it multiplies.
  • Check your solution by substituting it back into the original equation to see if it makes a true statement.
In the solution provided, once we had \(14 = \sqrt{4*a}\), we squared both sides to remove the square root and then divided by 4 to isolate \(a\), resulting in \(a = 49\).
Square Roots
The square root is a mathematical concept that gives us a number which, when multiplied by itself, equals the original number. Picture it as finding the side of a square when you know the area. If \(x\) is your original number, then the square root of \(x\) is represented as \(\sqrt{x}\). Whenever we deal with square roots, there are a few key points to remember:
  • The square root of a positive number is actually two numbers: a positive and a negative. For example, \(\sqrt{9} = 3\) and \(-3\).
  • Square roots are often used in formulas to solve problems involving areas, like when calculating geometric means.
  • In many cases, especially in real number contexts, we consider only the positive roots as solutions.
In the problem at hand, \(\sqrt{4*a} = 14\) means finding a number \(a\) such that when we multiply it by 4 and take the square root, the result is 14. Solving it involved reversing the square root process by squaring both sides, simplifying the equation so we could clearly find \(a = 49\).

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Most popular questions from this chapter

Does the equation model direct variation, inverse variation, or neither? \(x=\frac{7}{y}\)

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