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

Answer:

a = $\frac{1}{2}$. Surface Area = $\frac{4}{3}$$ft^{2}$. and area of the Dish = $\frac{4}{3}$$ft^{2}$+pi$4^{2}$ = $\frac{4}{3}$$ft^{2}$+50.27=51.6$ft^{2}$

Step-by-step 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 = $\frac{1}{2}$.

so we have now.

y(x) = $\frac{1}{2}x^{2}$ (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)$\frac{1}{32}$$x^{2}$]}]dy

Evaluating this integral gives. $\frac{4}{3}$$ft^{2}$.

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

= $\frac{4}{3}$$ft^{2}$+pi$4^{2}$ = $\frac{4}{3}$$ft^{2}$+50.27=51.6$ft^{2}$...

Note the Difference between area and aurface area.!