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 8-foot diameter and a maximum depth of 2 feet, find the value of a and the surface area (in square feet) of the dish. (Round the surface area to two decimal places.) What is the area and surface area?

Answer 1

a =
`(1)/(2)`. Surface Area =
`(4)/(3)`
`ft^(2)`. and area of the Dish =
`(4)/(3)`
`ft^(2)`+pi
`4^(2)` =
`(4)/(3)`
`ft^(2)`+50.27=51.6
`ft^(2)`

Explanation:

(1) Constant. y(x) = a
`x^(2)` that is the curve that we need to rotate around the y axis to get the parabola with diameter of 8 feet and 2 meter depth that statement is translated in mathematics as x = -4 to 4 and y = 0 to 2.

y max = 2, x max = 4 setting up a equation with a unknown gives

2=a4 and a =
`(1)/(2)`.

so we have now.

y(x) =
`(1)/(2)x^(2)` (Done with solving for a Constant).

(2) Surface Area.

Setting Up surface integral.

(i) range in x = 0 to 4.

(ii) range in y = 0 to 2.

integral is.

Integral(0-2)[{integral[(0-4)
`(1)/(32)`
`x^(2)`]}]dy

Evaluating this integral gives.
`(4)/(3)`
`ft^(2)`.

and area is surface area + area of the circle with 8ft diameter.

=
`(4)/(3)`
`ft^(2)`+pi
`4^(2)` =
`(4)/(3)`
`ft^(2)`+50.27=51.6
`ft^(2)`...

Note the Difference between area and aurface area.!