The antenna of a radio telescope is a paraboloid measuring 66 feet across with a depth of 35 feet. Determine, to the nearest tenth of a foot, the distance from the vertex to the focus of this antenna. Round to the nearest hundredth.
A)2.32 feet
B)9.28 feet
C)15.56 feet
D)31.11 feet
E)7.78 feet

Question

Quizplus · Accepted Answer

The formula for the focus of a parabola in the form $y = \frac{1}{4f}x^2$, where $f$ is the distance from the vertex to the focus, can be used here. Given the paraboloid's dimensions, we can use the depth (35 feet) as the value of $y$ and half the diameter (33 feet) as the value of $x$ to solve for $f$. Rearranging the formula to solve for $f$, we get $f = \frac{x^2}{4y}$. Substituting $x = 33$ and $y = 35$, we find $f = \frac{33^2}{4*35} = \frac{1089}{140} \approx 7.78$ feet.

The Antenna of a Radio Telescope Is a Paraboloid Measuring