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Chapter 5: Problem 80

Solve for the specified variable or expression. $$ A r-c^{2} d=d m^{2} \text { for } d $$

Short Answer

Expert verified
The solution for \(d\) is \(d = \frac{Ar}{m^2 + c^2}\).

Step by step solution

01

Isolate All Terms Involving d on One Side

To begin solving for the variable \(d\), we need to make sure that all terms involving \(d\) are on one side of the equation. Since both sides of the equation already contain \(d\), we will move the \(Ar-c^2d\) to the other side to get:\[ d m^2 + c^2 d = Ar \]
02

Factor Out d

Now that we have all the terms involving \(d\) on one side, we will factor \(d\) out of the terms on the left:\[ d(m^2 + c^2) = Ar \]
03

Solve for d

The final step is to solve for \(d\) by dividing both sides of the equation by \((m^2 + c^2)\):\[ d = \frac{Ar}{m^2 + c^2} \]

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Key Concepts

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

Isolating Variables
When solving equations, one key skill is to isolate the variable you are interested in. In this exercise, we're tasked with solving for the variable \( d \). Isolating a variable means getting this variable by itself on one side of the equal sign while moving everything else to the opposite side.
  • Start by identifying terms that include the variable of interest.
  • Move other terms to the opposing side of the equation by adding, subtracting, multiplying, or dividing as needed.
  • Ensure you perform the same operations to both sides of the equation to maintain equality.
In this exercise, we already have \( d \) terms on both sides, so the first move is to combine those terms by algebraic manipulation. We achieve this by shifting all terms with \( d \) to one side.
Factoring
Factoring is an essential step when multiple terms involve the variable you are solving for. Once you have isolated all the relevant terms, factoring helps simplify the equation.In our equation \( dm^2 + c^2d = Ar \), notice that \( d \) is common to both terms on the left-hand side:- Identify terms with a common factor, which in this case is \( d \).- Factor out these terms to consolidate them into one bracket.So, we extract \( d \), resulting in:\[ d(m^2 + c^2) = Ar \]This clever step transforms the complicated equation into a more manageable form, setting you up nicely for solving the isolated variable.
Algebraic Manipulation
Algebraic manipulation involves changing an equation鈥檚 form to reveal a desired result. After isolating the terms with \( d \) and factoring, the equation becomes simpler and more solvable.To find the solution for \( d \), we perform algebraic operations to shift other terms away from it. This technique ensures that the isolated \( d \) stands on its own:
  • Notice that \( d \) is multiplied by the expression \( (m^2 + c^2) \).
  • To solve for \( d \), divide both sides of the equation by \( (m^2 + c^2) \).
The algebraic manipulation allows us to neatly isolate \( d \):\[ d = \frac{Ar}{m^2 + c^2} \]This approach concludes the solution, showcasing the power of systematic algebraic steps in achieving the required form.

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Most popular questions from this chapter

Storage Tanks. The volume \(V(r)\) of the gasoline storage tank, in cubic feet, is given by the polynomial function \(V(r)=4.2 r^{3}+37.7 r^{2},\) where \(r\) is the radius in feet of the cylindrical part of the tank. What is the capacity of the tank if its radius is 4 feet? (PICTURE NOT COPY)

The following graphs of two polynomial functions \(f(x)=2 x^{3}-8 x\) and \(f(x)=2 x(x+2)(x-2)\) appear to be the same. After examining their equations, explain why we know that they are identical graphs. (Graph can't copy)

Factor each expression. $$ \frac{1}{36}-y^{4} $$

Explain why the terms \(5 x^{2} y\) and \(5 x y^{2}\) are not like terms.

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