Problem 123 - Elementary and Intermediate Algebra

Question

Use a calculator to solve each problem. Round answers to the nearest tenth. Embroidery. The radius \(r\) of a circle is given by the formula \(r=\sqrt{\frac{A}{\pi}},\) where \(A\) is its area. Find the diameter of the embroidery hoop if there are 38.5 in. \(^{2}\) of stretched fabric on which to embroider.

Step-by-Step Solution

Answer

The diameter of the embroidery hoop is 7.0 inches.

Step 1: Understand the Formula

The given formula to find the radius of a circle is \( r = \sqrt{\frac{A}{\pi}} \), where \( A \) is the area of the circle.

Step 2: Apply the Formula

Substitute \( A = 38.5 \) square inches into the formula: \( r = \sqrt{\frac{38.5}{\pi}} \).

Step 3: Calculate the Radius

Use a calculator to find the value of \( r \). First, compute the division: \( \frac{38.5}{\pi} \approx 12.243 \), and then find the square root: \( r \approx \sqrt{12.243} \approx 3.5 \) inches.

Step 4: Find the Diameter

The diameter \( D \) of a circle is twice the radius. Therefore, \( D = 2 \times r = 2 \times 3.5 = 7.0 \) inches.

Step 5: Round the Diameter

The diameter is already rounded to the nearest tenth, so \( D = 7.0 \) inches is the final answer.

Key Concepts

circle area formula

The circle area formula is all about connecting the dots between the area of a circle and other dimensions like the radius. The area of a circle is given by the formula:

• \( A = \pi r^2 \)

This formula expresses the total space covered inside the boundary of a circle. We often use pi (\( \pi \)) in calculations involving circles because it is the ratio of a circle's circumference to its diameter, roughly equal to 3.14159. This constant helps connect different circle properties harmoniously.
In reverse, if we're given the area and need to find the radius, we can rearrange the formula to solve for \( r \):

• \( r = \sqrt{\frac{A}{\pi}} \)

This particular formula allows us to understand how the area relates to the radius, which is crucial for further calculations, such as finding the diameter.

radius and diameter calculations

Calculating the radius and diameter involves understanding how these two components of a circle relate to each other. Knowing one can easily lead to calculating the other.
The radius of a circle is the distance from the center to any point on its boundary. It is a crucial measure because it directly influences the entire geometry of the circle. Once we have the radius, the diameter can be found quickly:

• The diameter \( D \) is always twice the radius:
  \( D = 2r \)

For example, if the radius is 3.5 inches, multiplying by two gives a diameter of 7.0 inches. Keep in mind, understanding the relationship between the radius and diameter is vital for problems related to circles.

geometry in algebra

Geometry often blends seamlessly with algebra, especially when dealing with circles. The concepts we've discussed manifest within many algebraic applications. One common intersection of these fields is when interpreting formulas where letters represent specific measurements or constants. For example, in our case:

• The letter \( A \) represents the area of the circle in the formula \( r = \sqrt{\frac{A}{\pi}} \)

Algebra is used to rearrange and solve these formulas based on the known quantities. Whether it’s finding the radius from the area or translating that into a diameter, these calculations show the essential skills of algebra meeting geometry.
Algebra provides the toolkit to isolate variables and solve for unknown measures, making complex geometric problems more manageable by breaking them down into formulaic expressions.