Answer :
We are given the circumference of the hub cap,
[tex]$$
C = 81.58 \text{ cm},
$$[/tex]
and the value of [tex]$\pi = 3.14$[/tex]. We want to find the area of the hub cap, as well as explain the effect of a smaller circumference on the area.
Step 1. Find the Radius
The circumference of a circle is related to its radius by the formula
[tex]$$
C = 2\pi r.
$$[/tex]
Solving for the radius [tex]$r$[/tex], we have
[tex]$$
r = \frac{C}{2\pi}.
$$[/tex]
Substitute the given values:
[tex]$$
r = \frac{81.58}{2 \times 3.14} \approx \frac{81.58}{6.28} \approx 12.99 \text{ cm}.
$$[/tex]
We round the intermediate value to the nearest thousandth, giving [tex]$r \approx 12.99$[/tex] cm.
Step 2. Calculate the Area
The area [tex]$A$[/tex] of a circle is given by the formula
[tex]$$
A = \pi r^2.
$$[/tex]
Now substitute [tex]$r \approx 12.99$[/tex] cm into the formula:
[tex]$$
A = 3.14 \times \left(12.99\right)^2.
$$[/tex]
Calculating the area:
[tex]$$
\left(12.99\right)^2 \approx 168.87,
$$[/tex]
so
[tex]$$
A \approx 3.14 \times 168.87 \approx 529.88 \text{ cm}^2.
$$[/tex]
Rounding [tex]$529.88$[/tex] to the nearest whole number, we get
[tex]$$
A \approx 530 \text{ cm}^2.
$$[/tex]
Step 3. Explain the Effect of a Smaller Circumference
If the circumference of the hub cap were smaller, the radius would also be smaller because the two are directly related by the equation
[tex]$$
r = \frac{C}{2\pi}.
$$[/tex]
A smaller radius means that when we compute the area
[tex]$$
A = \pi r^2,
$$[/tex]
the result will be significantly smaller, since the radius is squared in the area calculation. This shows that even a modest reduction in the circumference (and thus the radius) can lead to a much larger decrease in the area.
Final Answer:
The radius of the hub cap is approximately [tex]$12.99$[/tex] cm, and its area is about [tex]$530$[/tex] square centimeters. If the circumference were smaller, the area would decrease significantly because it depends on the square of the radius.
[tex]$$
C = 81.58 \text{ cm},
$$[/tex]
and the value of [tex]$\pi = 3.14$[/tex]. We want to find the area of the hub cap, as well as explain the effect of a smaller circumference on the area.
Step 1. Find the Radius
The circumference of a circle is related to its radius by the formula
[tex]$$
C = 2\pi r.
$$[/tex]
Solving for the radius [tex]$r$[/tex], we have
[tex]$$
r = \frac{C}{2\pi}.
$$[/tex]
Substitute the given values:
[tex]$$
r = \frac{81.58}{2 \times 3.14} \approx \frac{81.58}{6.28} \approx 12.99 \text{ cm}.
$$[/tex]
We round the intermediate value to the nearest thousandth, giving [tex]$r \approx 12.99$[/tex] cm.
Step 2. Calculate the Area
The area [tex]$A$[/tex] of a circle is given by the formula
[tex]$$
A = \pi r^2.
$$[/tex]
Now substitute [tex]$r \approx 12.99$[/tex] cm into the formula:
[tex]$$
A = 3.14 \times \left(12.99\right)^2.
$$[/tex]
Calculating the area:
[tex]$$
\left(12.99\right)^2 \approx 168.87,
$$[/tex]
so
[tex]$$
A \approx 3.14 \times 168.87 \approx 529.88 \text{ cm}^2.
$$[/tex]
Rounding [tex]$529.88$[/tex] to the nearest whole number, we get
[tex]$$
A \approx 530 \text{ cm}^2.
$$[/tex]
Step 3. Explain the Effect of a Smaller Circumference
If the circumference of the hub cap were smaller, the radius would also be smaller because the two are directly related by the equation
[tex]$$
r = \frac{C}{2\pi}.
$$[/tex]
A smaller radius means that when we compute the area
[tex]$$
A = \pi r^2,
$$[/tex]
the result will be significantly smaller, since the radius is squared in the area calculation. This shows that even a modest reduction in the circumference (and thus the radius) can lead to a much larger decrease in the area.
Final Answer:
The radius of the hub cap is approximately [tex]$12.99$[/tex] cm, and its area is about [tex]$530$[/tex] square centimeters. If the circumference were smaller, the area would decrease significantly because it depends on the square of the radius.
