Solving the Circular Field Area Ratio Problem

This problem involves the relationship between the areas of two circles and their radii. We are given the ratio of the areas of two circular fields and the radius of the larger field. We need to find the radius of the smaller field.

Understanding the Area of a Circle

The area of a circle is calculated using the formula:

\(\text{Area} = \pi r^2\)

where \(\pi\) (pi) is a mathematical constant approximately equal to 3.14159, and \(r\) is the radius of the circle.

Relating Area Ratio to Radius Ratio

Let the area of the smaller field be \(\text{Area}_s\) and its radius be \(r_s\). Let the area of the bigger field be \(\text{Area}_b\) and its radius be \(r_b\).

The ratio of the areas is given as 16 : 49. We can write this as:

\(\frac{\text{Area}_s}{\text{Area}_b} = \frac{16}{49}\)

Using the area formula, we can substitute \(\text{Area}_s = \pi r_s^2\) and \(\text{Area}_b = \pi r_b^2\) into the ratio equation:

\(\frac{\pi r_s^2}{\pi r_b^2} = \frac{16}{49}\)

Since \(\pi\) is a common factor in the numerator and denominator, it cancels out:

\(\frac{r_s^2}{r_b^2} = \frac{16}{49}\)

This can be written as:

\(\left(\frac{r_s}{r_b}\right)^2 = \frac{16}{49}\)

Calculating the Radius of the Smaller Field

To find the ratio of the radii, we take the square root of both sides of the equation:

\(\sqrt{\left(\frac{r_s}{r_b}\right)^2} = \sqrt{\frac{16}{49}}\)

\(\frac{r_s}{r_b} = \frac{\sqrt{16}}{\sqrt{49}}\)

\(\frac{r_s}{r_b} = \frac{4}{7}\)

We are given that the radius of the bigger field, \(r_b\), is 14 m.

Now we can substitute the value of \(r_b\) into the ratio equation:

\(\frac{r_s}{14} = \frac{4}{7}\)

To find \(r_s\), we multiply both sides of the equation by 14:

\(r_s = \frac{4}{7} \times 14\)

\(r_s = 4 \times \frac{14}{7}\)

\(r_s = 4 \times 2\)

\(r_s = 8\) m

Therefore, the radius of the smaller field is 8 m.

Summary of Steps

1. Identify the given ratio of areas: \(\text{Area}_s : \text{Area}_b = 16 : 49\).
2. Recall the formula for the area of a circle: \(\text{Area} = \pi r^2\).
3. Set up the ratio of areas using the radii: \(\frac{\pi r_s^2}{\pi r_b^2} = \frac{r_s^2}{r_b^2} = \left(\frac{r_s}{r_b}\right)^2\).
4. Equate the radius ratio squared to the area ratio: \(\left(\frac{r_s}{r_b}\right)^2 = \frac{16}{49}\).
5. Take the square root of both sides to find the ratio of radii: \(\frac{r_s}{r_b} = \frac{4}{7}\).
6. Substitute the given radius of the bigger field (\(r_b = 14\) m) into the equation.
7. Solve for the radius of the smaller field (\(r_s\)).

Revision Table: Circle Properties & Ratios

Additional Information: Solving Ratio Problems

Ratio problems often involve setting up proportions and using given values to find unknown quantities. When dealing with geometric shapes, understanding how different measurements (like radius, area, volume) scale with each other is crucial.

• For circles, the area scales with the square of the radius. This is why the ratio of areas is the square of the ratio of radii.
• If the ratio of radii were given, say \(r_1 : r_2 = a : b\), then the ratio of areas would be \(a^2 : b^2\). Conversely, if the ratio of areas is \(A_1 : A_2 = c : d\), then the ratio of radii is \(\sqrt{c} : \sqrt{d}\).
• Always make sure you are comparing corresponding quantities (e.g., radius to radius, area to area).
• Pay attention to whether the question asks for the ratio itself or an actual measurement based on a given value.

This type of problem reinforces the importance of understanding how formulas relate to geometric scaling.