o find the diameter of the larger circle
the circumference of the circle is 56.52 and the circumference of the smaller circle is 28.26 use a proportion to find the diameter of the larger circle
The Correct Answer and Explanation is:
To find the diameter of the larger circle using a proportion, we start with the relationship between circumference and diameter: Circumference=π×Diameter\text{Circumference} = \pi \times \text{Diameter}
We are given:
- Circumference of the larger circle: 56.52 units
- Circumference of the smaller circle: 28.26 units
Let’s assume:
- Diameter of the smaller circle = d₁
- Diameter of the larger circle = d₂
From the formula above: Circumference of larger circleCircumference of smaller circle=d2d1\frac{\text{Circumference of larger circle}}{\text{Circumference of smaller circle}} = \frac{d_2}{d_1}
Substitute the given values: 56.5228.26=d2d1\frac{56.52}{28.26} = \frac{d_2}{d_1}
Now simplify: 2=d2d12 = \frac{d_2}{d_1}
This tells us that the diameter of the larger circle is twice that of the smaller circle.
To find the actual diameter of the larger circle, we first compute the diameter of the smaller circle using: Circumference=π×Diameter⇒Diameter=Circumferenceπ\text{Circumference} = \pi \times \text{Diameter} \Rightarrow \text{Diameter} = \frac{\text{Circumference}}{\pi}
So, d1=28.26π≈28.263.1416≈9d_1 = \frac{28.26}{\pi} \approx \frac{28.26}{3.1416} \approx 9
Therefore, d2=2×d1=2×9=18d_2 = 2 \times d_1 = 2 \times 9 = 18
Final Answer:
The diameter of the larger circle is 18 units.
Explanation
This problem is based on understanding proportional relationships between the dimensions of circles. The key geometric formula at play is the relationship between a circle’s circumference (C) and its diameter (d), given by the formula: C=π×dC = \pi \times d
Since both circles are true circles, their circumferences and diameters will maintain a consistent ratio. This means that the ratio of their circumferences is equal to the ratio of their diameters.
We’re told the larger circle has a circumference of 56.52 units and the smaller one 28.26 units. By dividing these, we get: 56.5228.26=2\frac{56.52}{28.26} = 2
This tells us the larger circle’s diameter is twice the smaller’s. To determine the actual size, we calculate the smaller circle’s diameter by dividing its circumference by π (approximately 3.1416), which gives us: 28.263.1416≈9 units\frac{28.26}{3.1416} \approx 9 \text{ units}
Multiplying by 2 gives us the larger circle’s diameter: 9×2=18 units9 \times 2 = 18 \text{ units}
This approach shows how proportional reasoning, along with understanding formulas involving π, allows us to find unknown dimensions. It’s a practical application of algebra and geometry working together.
