Question 11.2.4: Calculating Some Properties of the Hubble Space Telescope Th...

Position the parabola so that its vertex x is at the origin and its focus is on the positive y-axis. The equation of the parabola is of the form

\begin{aligned}&x^{2}=4 a y\\&x^{2}=4(2304) y \quad a=2304\\&x^{2}=9216 y \quad \text { Simplify. }\end{aligned}

To find the thickness y of the mirror at the edge, substitute x = 47.25 (half the diameter) in the equation x^{2}=9216 y and solve for y.

\begin{array}{llrl}(47.25)^{2} & =9216 y & & \text { Replace } x \text { with } 47.25 \\\frac{(47.25)^{2}}{9216} & =y & & \text { Divide both sides by } 9216 \\0.242248 & \approx y & & \text { Use a calculator. }\end{array}

Thus, the thickness of the mirror at the edges is approximately 0.242248 inch.
The Hubble telescope initially had “blurred vision” because the mirror was ground at the edges to a thickness of 0.24224 inch. In other words, it had been ground 0.000008 inch, or two-millionths of a meter, smaller than it should have been! The problem was fixed in 1993.