Respuesta :
[tex]1808.64 inches[/tex] much rope with the gym teacher need .
Step-by-step explanation:
Here we have , In the gym there is a round metal pole that is 8 inches in diameter. The gym teacher wants to wrap the pole tightly with 1-inch thick rope from the ground up to a height of 6 feet. (The rope will be wound over and over until it reaches a height of 6 feet.) We need to find About how much rope with the gym teacher need . Let's find out:
According to question , the rope is covered in lateral surface area of a cylinder i.e. to wrap the pole tightly with 1-inch thick rope from the ground up to a height of 6 feet which will cover lateral surface area of cylinder . We know that Formula for lateral surface area is :
⇒ [tex]2\pi r h[/tex]
From the question we have
[tex]Radius = \frac{Diameter}{2} =\frac{8}{2} =4 inches\\Height = 6 (1feet) = 6(12inches)=72inches[/tex]
Now , height of pole is 72 inches and we will wrap pole with 1 inch thick rope , So total number of turns of rope in pole :
⇒ [tex]\frac{72}{1}[/tex]
⇒ [tex]72 Turns[/tex]
Now , Length of one turn of rope in pole is equivalent to circumference of circle with radius = 4 inches as:
⇒ [tex]Length = Circumference = 2\pi r[/tex]
⇒ [tex]Length = 2\pi (4)[/tex]
⇒ [tex]Length =25.12inches[/tex]
Since , length of one turn is 25.12 inch so ,length of total rope i.e. of 72 turns is :
⇒ [tex]Length = 72(25.12)[/tex]
⇒ [tex]Length = 1808.64inches[/tex]
Therefore , [tex]1808.64 inches[/tex] much rope with the gym teacher need .
