Question

a gardener makes a new circular flower bed. the bed is ten feet in diameter. calculate the circumference and the area of the circular flower bed.
circumference = 5π feet, area = 25π square feet
circumference = 10π feet, area = 100π square feet
circumference = 10π feet, area = 25π square feet
circumference = 10 feet, area = 10π square feet

Explanation:

Step1: Recall circle formulas

The formula for the circumference of a circle is \( C = \pi d \) (where \( d \) is the diameter) or \( C = 2\pi r \) (where \( r \) is the radius). The formula for the area of a circle is \( A=\pi r^{2}\). Given the diameter \( d = 10\) feet, the radius \( r=\frac{d}{2}=\frac{10}{2} = 5\) feet.

Step2: Calculate circumference

Using \( C=\pi d \), substitute \( d = 10\): \( C=\pi\times10=10\pi \) feet. Wait, no, wait: Wait, if \( d = 10\), then \( C=\pi d=10\pi \)? Wait, no, wait the first option has circumference \( 5\pi \), but let's re - check. Wait, diameter \( d = 10\), so radius \( r = 5\). Circumference \( C = 2\pi r=2\pi\times5 = 10\pi \)? Wait, no, \( C=\pi d\), \( d = 10\), so \( C = 10\pi \)? But the first option says circumference \( 5\pi \). Wait, maybe I made a mistake. Wait, no, the diameter is 10, so radius is 5. Circumference \( C=2\pi r = 10\pi \), area \( A=\pi r^{2}=\pi\times5^{2}=25\pi \). Wait, let's check the options:

Option 1: circumference \( 5\pi \), area \( 25\pi \) – wrong, because \( C=\pi d = 10\pi \) (since \( d = 10\)).

Option 2: circumference \( 10\pi \), area \( 100\pi \) – wrong, area should be \( 25\pi \).

Option 3: circumference \( 10\pi \), area \( 25\pi \) – let's check: \( C=\pi d=10\pi \) (d = 10), \( r=\frac{d}{2}=5 \), \( A=\pi r^{2}=\pi\times25 = 25\pi \). Yes, this is correct.

Option 4: circumference \( 10 \), area \( 10\pi \) – wrong, circumference formula is \( \pi d \) or \( 2\pi r \), so it should have \( \pi \), and area is \( \pi r^{2}\).

Wait, maybe the options are:

1. circumference \( = 5\pi \) feet, area \( = 25\pi \) square feet

1. circumference \( = 10\pi \) feet, area \( = 100\pi \) square feet

1. circumference \( = 10\pi \) feet, area \( = 25\pi \) square feet

1. circumference \( = 10 \) feet, area \( = 10\pi \) square feet

So the correct one is the third option (the one with circumference \( 10\pi \) feet and area \( 25\pi \) square feet). Wait, but let's do the calculation properly:

Given diameter \( d = 10\) feet.

Step 1: Calculate the circumference

The formula for the circumference of a circle is \( C=\pi d\) (where \( d \) is the diameter) or \( C = 2\pi r\) (where \( r \) is the radius). Since \( d = 10\) feet, using \( C=\pi d\), we get \( C=\pi\times10=10\pi \) feet.

Step 2: Calculate the radius

The radius \( r\) of a circle is half of the diameter, so \( r=\frac{d}{2}=\frac{10}{2}=5\) feet.

Step 3: Calculate the area

The formula for the area of a circle is \( A = \pi r^{2}\). Substituting \( r = 5\) into the formula, we get \( A=\pi\times5^{2}=\pi\times25 = 25\pi \) square feet.

Answer:

The option with "circumference \( = 10\pi \) feet, area \( = 25\pi \) square feet" (the third option among the given choices)