The areas of two circular fields are in the ratio 16 ∶ 49. If the radius of the bigger field is 14 m then what is the radius of the smaller field?
8 m
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
- Identify the given ratio of areas: \(\text{Area}_s : \text{Area}_b = 16 : 49\).
- Recall the formula for the area of a circle: \(\text{Area} = \pi r^2\).
- Set up the ratio of areas using the radii: \(\frac{\pi r_s^2}{\pi r_b^2} = \frac{r_s^2}{r_b^2} = \left(\frac{r_s}{r_b}\right)^2\).
- Equate the radius ratio squared to the area ratio: \(\left(\frac{r_s}{r_b}\right)^2 = \frac{16}{49}\).
- Take the square root of both sides to find the ratio of radii: \(\frac{r_s}{r_b} = \frac{4}{7}\).
- Substitute the given radius of the bigger field (\(r_b = 14\) m) into the equation.
- Solve for the radius of the smaller field (\(r_s\)).
| Quantity | Smaller Field | Bigger Field |
|---|---|---|
| Area Ratio | 16 | 49 |
| Radius Ratio (\(\sqrt{\text{Area Ratio}}\)) | \(\sqrt{16} = 4\) | \(\sqrt{49} = 7\) |
| Actual Radius | \(r_s\) (unknown) | 14 m |
| Relationship | \(\frac{r_s}{r_b} = \frac{4}{7}\) | |
| Calculation | \(r_s = \frac{4}{7} \times r_b = \frac{4}{7} \times 14 = 8\) m | |
Revision Table: Circle Properties & Ratios
| Concept | Formula/Relationship | Notes |
|---|---|---|
| Area of Circle | \(\text{Area} = \pi r^2\) | \(r\) is the radius |
| Circumference of Circle | \(\text{Circumference} = 2\pi r\) or \(\pi d\) | \(d\) is the diameter |
| Ratio of Areas of Two Circles | \(\frac{\text{Area}_1}{\text{Area}_2} = \frac{\pi r_1^2}{\pi r_2^2} = \left(\frac{r_1}{r_2}\right)^2\) | Ratio of areas is the square of the ratio of radii |
| Ratio of Radii of Two Circles | \(\frac{r_1}{r_2} = \sqrt{\frac{\text{Area}_1}{\text{Area}_2}}\) | Ratio of radii is the square root of the ratio of areas |
| Ratio of Circumferences of Two Circles | \(\frac{\text{Circumference}_1}{\text{Circumference}_2} = \frac{2\pi r_1}{2\pi r_2} = \frac{r_1}{r_2}\) | Ratio of circumferences is the same as the ratio of radii |
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.
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