Question

A spiral staircase turns as it rises 10 feet. The radius of the staircase is 3 feet. What is the number of feet in the length of the handrail

Answer 1

The number of feet in the length of the handrail is 17.3 feet.

What is a Pythagorean Theorem?

Pythagorean theorem, is a geometric theorem that the total of a squares on the sides of a right triangle equals the square upon that hypotenuse (a side opposite its right angle)—or, using familiar algebraic notation, the sum of squared on the hypotenuse equals the square also on hypotenuse.

H² = B² + P²

A spiral would be a shape that wraps around and around with each curve being higher or lower than the one before it.

Now, according to the question;

The simplest way to understand this is to assume that we may "unfold" this spiral to form a rectangle.

Its bottom part of rectangle is simply an arc length of a 3 ft radius circle spun around 270°= (3/4) of a full revolution.

Thus,

= (3/4)× (2$\pi$) ×(3)  

=  (9/2)×$\pi$ ft

=   4.5×$\pi$  ft

The height of the staircase will be the side of rectangle = 10ft.

And the rectangle's diagonal will correspond to the length of a railing.

To solve this, we can apply the Pythagorean Theorem.

Handrail length = √ [ (4.5 ×$\pi$)² + 10² ]  

Handrail length ≈  17.3 ft

Therefore, the the number of feet in the length of the handrail is 17.3 feet.

To know more about Pythagorean Theorem, here

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The complete question is-

A spiral staircase turns 270° as it rises 10 feet. The radius of the staircase is 3 feet. What is the number of feet in the length of the handrail?