Answer
$3$ ft.
Work Step by Step
Step 1. Using the given function $x^2+y^2=225$ for the circle, we have $2xdx+2ydy=0$ and $\frac{dy}{dx}=-\frac{x}{y}$ which gives the slopes of the tangent lines to the circle.
Step 2. At point $(12,-9)$, the slope is $m=-\frac{12}{-9}=\frac{4}{3}$ and the equation for the tangent line is $y+9=\frac{4}{3}(x-12)$
Step 3. Half of the width of the Gondola is given by the x-coordinate on the tangent line when $y=-23$ft as given in the figure of the Exercise. We have $-23+9=\frac{4}{3}(x-12)$, which gives $x=12-\frac{3}{4}(14)=\frac{3}{2}$
Step 4. The width of the Gondola is then $3$ ft.
