Question

In If the radius of a circle is 15 inches and the diameter is \((3 x+7)\) inches, what is the value of \(x\) ? Round to the nearest hundredth.

Short answer

The value of \(x\) is 7.67.

Step-by-step solution

Understand the Relationship Between Diameter and Radius

The diameter of a circle is twice the radius. Given that the radius is 15 inches, the diameter can be calculated using the formula: \( \text{Diameter} = 2 \times \text{Radius} \). This gives us: \( \text{Diameter} = 2 \times 15 = 30 \) inches.

Set Up the Equation

We know from the problem that the diameter can also be expressed as \( 3x + 7 \) inches. Since we have calculated the diameter as 30 inches, we can set up the equation: \( 3x + 7 = 30 \).

Solve for \(x\)

To find \(x\), first subtract 7 from both sides of the equation to isolate the term with \(x\):\[ 3x + 7 - 7 = 30 - 7 \]This simplifies to: \( 3x = 23 \).Next, divide both sides by 3 to solve for \(x\):\[ x = \frac{23}{3} \approx 7.67 \].

Round the Answer

The value of \(x\) obtained is approximately 7.67. According to the problem, we need to round this to the nearest hundredth. Therefore, the value of \(x\) is approximately 7.67.

Key concepts

Radius and Diameter Relationship

The relationship between the radius and diameter is important when dealing with circles. The diameter is simply twice the radius. This means if you know either the radius or the diameter of a circle, you can easily find the other. For example, if the radius of a circle is 15 inches, the diameter would be calculated as follows:

• Calculate by multiplying the radius by 2: \[ \text{Diameter} = 2 \times \text{Radius} = 2 \times 15 = 30 \text{ inches} \]

Understanding this relationship helps solve many geometry problems involving circles. By using the formula \( \text{Diameter} = 2 \times \text{Radius} \), you can find one when given the other with ease.
This basic concept is a stepping stone for solving more complex problems that involve setting up equations.

Equation Solving

Solving equations is a fundamental skill in algebra. It involves finding the variable's value that makes an equation true. In the given problem, we're tasked with finding the value of \(x\) in the equation:

• \[3x + 7 = 30\]

To solve this, we start by isolating the term with \(x\):

• Subtract 7 from both sides: \[3x + 7 - 7 = 30 - 7\]
• Simplify it to: \[3x = 23\]

Next, divide both sides by 3 to solve for \(x\):

• \[x = \frac{23}{3} \]

This type of step-by-step method is useful in breaking down complex problems into more manageable parts.
It is about systematically isolating the variable to find the solution, using basic arithmetic operations.

Rounding Numbers

Rounding numbers helps simplify them for ease of understanding or when a precise figure is not necessary. Let's focus on rounding to the nearest hundredth, which means rounding to the second decimal place. When we have:

• \(x \approx 7.67\)

To determine if a number needs to be rounded up or stays the same, look at the third decimal place. If it's 5 or more, round up.

• Since 7.67 has no third decimal place other than a zero, there's nothing to round, thus it stays 7.67.

Rounding is handy not just in math, but in everyday contexts as well.
It ensures that numbers are easy to communicate and utilize, especially when precision isn't paramount, like when interpreting results or making quick calculations.