Answer :
Final answer:
The area of the hub cap, modeled as a circle with a given circumference of 82.77 cm, is approximately 545.45 square centimeters. If the circumference were smaller, the area would decrease significantly because the area is proportional to the square of the radius.
Explanation:
To find the area of the hub cap, which we understand to be a circle, we first need to find the radius of the circle. The circumference (C) of a circle is related to its radius (r) by the formula C = 2πr, where π is a constant approximately equal to 3.14.
Given that the circumference is 82.77 centimeters, we can rearrange the formula to solve for the radius:
r = C / (2 π ).
Now substituting the provided values:
- r = 82.77 cm / (2 X 3.14)
- r = 82.77 cm / 6.28
- r = 13.18 cm (rounded to two decimal places)
With the radius calculated, we can then find the area (A) of the circle using the formula
A = π r^2.
Substituting the calculated radius into the area formula:
- A = 3.14 (13.18 cm)^2
- A = 3.14 X 173.71 cm^2
- A = 545.45 cm^2 (rounded to two decimal places)
The area of the hub cap is approximately 545.45 square centimeters.
If the circumference of the hub cap were smaller, the radius would also be smaller. Since the area of a circle depends on the square of its radius, even a small reduction in the circumference (and therefore the radius) would result in a relatively larger decrease in the area of the hub cap. This is because the area of a circle changes with the square of the radius.
