Answer:
a) $497.70
b) 392.4 square feet
c) $1,200
Step-by-step explanation:
A regular octagon has 8 sides of equal length.
Given each side of the octagon measures 9 feet in length, and one side does not have a railing, the total length of the railing is 7 times the length of one side:
[tex]\textsf{Total length of railing}=\sf 7 \times 9\; ft=63\;ft[/tex]
If the railing sells for $7.90 per foot, the total cost of the railing can be calculated by multiplying the total length by the cost per foot:
[tex]\textsf{Total cost of railing}=\sf 63\;ft \times \dfrac{\$7.90}{ft}=\$497.70[/tex]
Therefore, the cost of the railing is $497.70.
[tex]\hrulefill[/tex]
To find the area of the regular octagonal gazebo, given the side length and apothem, we can use the area of a regular polygon formula:
[tex]\boxed{\begin{minipage}{6cm}\underline{Area of a regular polygon}\\\\$A=\dfrac{n\;s\;a}{2}$\\\\where:\\\phantom{ww}$\bullet$ $n$ is the number of sides.\\ \phantom{ww}$\bullet$ $s$ is the length of one side.\\ \phantom{ww}$\bullet$ $a$ is the apothem.\\\end{minipage}}[/tex]
Substitute n = 8, s = 9, and a = 10.9 into the formula and solve for A:
[tex]\begin{aligned}\textsf{Area of the gazebo}&=\sf \dfrac{8 \times 9\:ft \times10.9\:ft}{2}\\\\&=\sf \dfrac{784.8\;ft^2}{2}\\\\&=\sf 392.4\; \sf ft^2\end{aligned}[/tex]
Therefore, the area of the gazebo is 392.4 square feet.
[tex]\hrulefill[/tex]
To calculate the cost of the gazebo's flooring if it costs $3 square foot, multiply the area of the gazebo found in part (b) by the cost per square foot:
[tex]\begin{aligned}\textsf{Total cost of flooring}&=\sf 392.4\; ft^2 \times \dfrac{\$3}{ft^2}\\&=\sf \$1177.2\\&=\sf \$1200\; (nearest\;hundred\;dollars)\end{aligned}[/tex]
Therefore, the cost of the gazebo's flooring to the nearest hundred dollars is $1,200.
a. To find the perimeter of the gazebo, we can use the formula P = 8s, where s is the length of one side. Substituting s = 9, we get:
P = 8s = 8(9) = 72 feet
Since one side is open, we only need to find the cost of railing for 7 sides. Multiplying the perimeter by 7, we get:
Cost = 7P($7.90/foot) = 7(72 feet)($7.90/foot) = $4,939.20
Therefore, the cost of the railing is $4,939.20.
b. To find the area of the gazebo, we can use the formula A = (1/2)ap, where a is the apothem and p is the perimeter. Substituting a = 10.9 and p = 72, we get:
A = (1/2)(10.9)(72) = 394.56 square feet
Therefore, the area of the gazebo is 394.56 square feet.
c. To find the cost of the flooring, we need to multiply the area by the cost per square foot. Substituting A = 394.56 and the cost per square foot as $3, we get:
Cost = A($3/square foot) = 394.56($3/square foot) = $1,183.68
Rounding to the nearest hundred dollars, the cost of the flooring is $1,184. Therefore, the cost of the gazebo's flooring is $1,184.
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