A Cylindrical silo is 37 ft. high and has a diameter of 14 ft. The top of the silo can be reached by a spiral staircase that circles the silo once. What is the angle of inclination of the staircase?

Guest Oct 27, 2015

#2**+5 **

Hi Rom and guest,

I agree this is an interesting question but I would do it differently, Maybe my way is wrong.

Rom I don't think that you found the angle did you?

The staircase goes around the silo much like the label goes around a can of peas

If you take the label off the peas, it is a rectangle.

The length will be the circumference of the can 2*pi*r

and the height will be the hieght of the can.

so

getting back to the silo, the stairs go from the bottom corner of the 'rectangle' to the diagonally opposite corner.

like this.

I wanted to draw the diagram but Geogebra has stopped working for me

Anyway it is a rectangle. and the stairs are the diagonal.

The rectangle is 37 foot hight and 2*pi*7 = 14pi foot long

The incline angle is the angle between the base and the diagonal.

\(tan\theta = \frac{37}{14\pi}\\ \theta = atan\left( \frac{37}{14\pi} \right)\\ \theta = 40^04'\)

So the angle of incline is 40.072 degrees (3dp)

or 40 degrees and 4 minutes to the nearest minute

Melody
Oct 27, 2015

#1**+5 **

Interesting problem.

We treat the staircase as a space curve.

\(r(t)=\{7 \cos(2 \pi t), 7 \sin(2 \pi t), 37 t\},~~~~0\leq t \leq 1\)

To find the length of the curve we integrate the norm of the time derivative of the curve.

\(L = \displaystyle{\int_0^1}\|\dot{r}\|~dt\)

\(\dot{r}=\{-14\pi \sin(2 \pi t), 14\pi \cos(2\pi t), 37\}\\ \|\dot{r}\| = \sqrt{(14\pi)^2+37^2} \\ L=\displaystyle{\int_0^1}\sqrt{(14\pi)^2+37^2}~dt = \sqrt{(14\pi)^2+37^2}\approx 57.48 ft\)

Rom
Oct 27, 2015

#2**+5 **

Best Answer

Hi Rom and guest,

I agree this is an interesting question but I would do it differently, Maybe my way is wrong.

Rom I don't think that you found the angle did you?

The staircase goes around the silo much like the label goes around a can of peas

If you take the label off the peas, it is a rectangle.

The length will be the circumference of the can 2*pi*r

and the height will be the hieght of the can.

so

getting back to the silo, the stairs go from the bottom corner of the 'rectangle' to the diagonally opposite corner.

like this.

I wanted to draw the diagram but Geogebra has stopped working for me

Anyway it is a rectangle. and the stairs are the diagonal.

The rectangle is 37 foot hight and 2*pi*7 = 14pi foot long

The incline angle is the angle between the base and the diagonal.

\(tan\theta = \frac{37}{14\pi}\\ \theta = atan\left( \frac{37}{14\pi} \right)\\ \theta = 40^04'\)

So the angle of incline is 40.072 degrees (3dp)

or 40 degrees and 4 minutes to the nearest minute

Melody
Oct 27, 2015

#3**+5 **

Here's a simpler solution:

Imagine that the silo is a"tube" which we can roll out into a flat plane.....assuming that the staircase makes only one complete turn from the bottom to the top of the silo.......we will have a rectangle that will be 37 ft high and will have a width of pi*d = pi* 14 ≈ 43.98 ft......and the staircase will be the diagonal of this rectangle....and the angle of inclination will be given by:

tan^{-1} (37 / 43.98) = about 40.07°

CPhill
Oct 27, 2015