Question

Solve for \(x\) in the following proportions. Carry division two decimal places as necessary. \(\frac{60}{1}=\frac{x}{2^{1 / 4}}\)

Short answer

The value of \( x \) is 71.40.

Step-by-step solution

Understanding the Proportion

The equation given is \( \frac{60}{1} = \frac{x}{2^{1/4}} \). This is a proportion, where two ratios are equal. Our goal is to solve for \( x \).

Cross-Multiplication

To solve for \( x \), we cross-multiply. This means multiplying the numerator of one fraction by the denominator of the other. Set up the equation: \( 60 \times 2^{1/4} = 1 \times x \), which simplifies to \( x = 60 \times 2^{1/4} \).

Calculating the Power

We need to calculate \( 2^{1/4} \). This is the fourth root of 2. The approximate value of \( 2^{1/4} \) is 1.189207. We round this value to two decimal places, which gives us 1.19.

Substitution and Final Calculation

Substitute \( 1.19 \) for \( 2^{1/4} \) in the equation \( x = 60 \times 2^{1/4} \). This gives \( x = 60 \times 1.19 \).

Performing the Multiplication

Calculate \( 60 \times 1.19 \). Multiply to get \( x = 71.40 \).

Key concepts

Cross-Multiplication

Cross-multiplication is a powerful technique used to solve proportion problems. A proportion is an equation where two ratios or fractions are set equal to each other, like in the equation \( \frac{a}{b} = \frac{c}{d} \). To find the unknown variable, we use cross-multiplication to get rid of the fractions. The basic idea is to multiply the numerator of one fraction by the denominator of the other fraction and set these two products equal.

For instance, if you have \( \frac{60}{1} = \frac{x}{2^{1/4}} \), cross-multiplication involves multiplying 60 by \( 2^{1/4} \) and 1 by \( x \), resulting in \( 60 \times 2^{1/4} = x \). This method effectively turns the proportion into a simple equation that is easier to solve. Here are some key steps:

• Identify the numerator and denominator in both fractions.
• Multiply the numerator of the first fraction by the denominator of the second.
• Multiply the numerator of the second fraction by the denominator of the first.
• Set the two products equal to form an equation.

Tackling proportions with cross-multiplication simplifies the path to finding unknowns.

Calculating Roots

To solve equations involving powers, like \( 2^{1/4} \), we need to calculate roots. In this exercise, we're dealing with finding the fourth root of 2. Roots are the inverse operation of raising numbers to a power.

In simpler terms, calculating \( 2^{1/4} \) means asking "What number, when raised to the power of 4, equals 2?" Fortunately, technology and calculators make finding these values easier. For instance:

• You can use a scientific calculator to directly compute \( 2^{1/4} \).
• Alternatively, you can look up the value in tables or computational tools.

In this case, \( 2^{1/4} \) approximates to 1.189207. For practical purposes, especially when instructed to round, we use 1.19.

Having a good understanding of how to compute and use roots and powers is crucial when working with proportions and many other areas in algebra.

Solving Equations

Solving equations is all about finding the value of the unknown variable that makes the equation true. This is a fundamental skill in algebra and mathematics as a whole.

Once we have set up our equation from cross-multiplication, like \( x = 60 \times 2^{1/4} \), we can easily substitute the calculated value of \( 2^{1/4} \) to find \( x \). Let's break it down:

• First, calculate or substitute any powers or roots needed in the equation.
• If you calculated \( 2^{1/4} \) to be approximately 1.19, substitute that into the equation to simplify it.
• Perform necessary arithmetic operations, here, multiplying 60 by 1.19.
• The result gives the value of the unknown variable \( x = 71.40 \).

By following these steps, solving the equation becomes a straightforward calculation. Simplifying each part step-by-step leads to clear, correct results. Understanding how to efficiently manipulate algebraic expressions and values is key to solving any equation you encounter.