First, let's find the height of the cylinder, using the Pythagorean theorem. Since the radius of the cylinder is 11 ft, we have a right triangle with legs of length 11 and 12, and hypotenuse 11 + 12 = 23. Therefore, the height of the cylinder is $\sqrt{23^2 - 11^2} = 20$.
Now we know that silos have cross-sectional radius 11 ft and height 20 ft.
Let $(a,b)$ be the tangency point where line of sight #1 meets the cylinder. Then by the Pythagorean theorem, $a^2 + b^2 = 11^2$, so $b^2 = 121 - a^2$. Also, the slope of the line of sight is $\frac{b}{a + 12}$. Let's find the slope of the radial line through $(a,b)$, which has an equation in the form of $y=kx$. The radial line goes through the origin $(0,0)$ and the tangential point $(a,b)$, so we have $b = ka$ and $k = \frac{b}{a}$. Since the line of sight and the radial line are perpendicular, their slopes multiply to -1, so we have:
$\frac{b}{a+12} \cdot \frac{b}{a} = -1$
Substitute $b^2 = 121 - a^2$ into the equation:
$\frac{121 - a^2}{a(a+12)} = -1$
Now solve for $a$:
$121 - a^2 = -a^2 - 12a$
$12a = 121$
$a \approx 10.083$
Now we can find $b$:
$b^2 = 121 - a^2 \approx 121 - 100.667$
$b \approx 4.514$
To find the portion of the y-axis that Dave cannot see, we need to find the y-coordinates of the tangency points on the silo. For line of sight #1, the y-coordinate of the tangency point is $\approx 4.514$. For line of sight #2, we notice that the tangency point will have the same y-coordinate value, but in the negative direction, so it is $\approx -4.514$. Therefore, the portion of the y-axis that Dave cannot see is the interval:
$\boxed{[-4.514, 4.514]}$ (to three decimal places)
Dave is going to leave academia and go into business building grain silos. A grain silo is a cylinder with a hemispherical top, used to store grain for farm animals. Here is a 3D view, a cross-section, and the top view.
If Dave is standing next to a silo of cross-sectional radius r = 11 feet at the indicated position, his vision will be partially obstructed. Find the portion of the y-axis that Dave cannot see. (Hint: Let a be the x-coordinate of the point where line of sight #1 is tangent to the silo; compute the slope of the line using two points (the tangent point and (12, 0)). On the other hand, compute the slope of line of sight #1 by noting it is perpendicular to a radial line through the tangency point. Set these two calculations of the slope equal and solve for a. Enter your answer using interval notation. Round your answer to three decimal places.)
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