Answer
Area of deck = big circle (pool + deck) - small circle (pool) = 69$\pi$$ft^{2}$
Amount of fence required to enclose the fence = circumference of big circle (pool + deck) = 26$\pi$ft
Work Step by Step
What is the area of the deck?
We can see that the area of the deck = area of big circle (pool + deck) - area of small circle (pool)
SOLVING FOR THE AREA OF THE SMALL CIRCLE:
Recall that the area of a circle is equal to $\pi$$r^{2}$, where r is the radius of the circle
The diameter of the small circle is 20ft as shown in the diagram.
We can divide the diameter of the circle by 2 to get the radius.
Radius of little circle
= 20ft $\div$ 2
= 10ft
Know that we know the radius, we can find the area
Area of little circle
= $\pi$$r^{2}$
= $\pi$$(10ft)^{2}$
= 100$\pi$$ft^{2}$
The area of the small circle is 100$\pi$$ft^{2}$
SOLVING FOR THE AREA OF THE BIG CIRCLE:
The radius of the big circle is equal to the radius of the small circle + the width of the deck
We know that the deck has a width of 3ft
We also know from before that the radius of the small circle is equal to 10ft
radius of big circle
= radius of small circle + width of deck
= 10ft + 3ft
= 13ft
Now that we know the radius, we can substitute the radius into the equation Area = $\pi$$r^{2}$
Area of big circle
= $\pi$$r^{2}$
= $\pi$$(13ft)^{2}$
= 169$\pi$$ft^{2}$
The area of the big circle is 169$\pi$$ft^{2}$
SOLVING FOR THE AREA OF THE DECK
Area of deck
= area of big circle (pool + deck) - area of small circle (pool)
= 169$\pi$$ft^{2}$ - 100$\pi$$ft^{2}$
= 69$\pi$$ft^{2}$
The area of the deck is 69$\pi$$ft^{2}$
How much fence is required to enclose the deck?
The amount of fence required to enclose the deck is equal to the circumference of the big circle
Recall that the circumference of a circle is equal to $\pi$d or 2$\pi$r, where r is the radius and d is the diameter
Since we already know from before that the radius of the big circle is 13$\pi$, we can plug this number into the equation for circumference = 2$\pi$r
Circumference of big circle
= 2$\pi$r
= (2$\pi$)(13ft)
= 26$\pi$
The amount of fence required to enclose the deck is 26$\pi$ft
