Question

A group of engineers is building a parabolic satellite dish whose shape will be formed by rotating the curve y = ax 2 about the y-axis. If the dish is to have a 18-ft diameter and a maximum depth of 4 ft, find the value of a and the surface of area of the dish.

Select the correct answer.

Answer 1

i dunno

Answer 2

any ideas zed?

Answer 3

i'm just having a look, trying to see if the answer i get is one of the options. :) give me some time

Answer 4

mk, thanks

Answer 5

I guess you don't know what x or y is? I'm thinking that x is the radius of the dish.

Answer 6

for the value of a

we need to have a depth of 4ft and a diameter of 18ft

so when x=9, y=4
4=a*9^2
a=4/81

i'm still working on the surface area :)

Answer 7

mk, thanks

Answer 8

For the S.A the rule to rotate about the y-axis is

\[S=\int 2\pi x ds\] where \[ds=\sqrt{1+(\frac{dy}{dx})^2dx}\] for if y=f(x) \(c\le x\le d\)

y=\(\frac{4}{81}\)x^2

Our limits are for (y=0 and y=4) x=0 and x=9

\[ds=\sqrt{1+(\frac{dy}{dx})^2dx}\]\[=\sqrt{1+(\frac{8}{81}x)^2dx}\]\[=\sqrt{1+\frac{64}{6461}x^2dx}\]

\[S=\int^9_0 2\pi x \sqrt{1+\frac{64}{6461}x^2dx}\]\[= (\frac{\pi(64x^2+6561)^{3/2}}{7776}) ^9_0\]\[= \frac{3}{32}(145\sqrt{145}-729)\pi\]\[\approx 299.54\]

I can't seem to get the answer. Maybe I've made some silly error but this is the method I used with the website below.

http://tutorial.math.lamar.edu/Classes/CalcII/SurfaceArea.aspx