Question

Here is a schematic drawing of the 90− ft dome used by the U.S. National Weather Service to house radar in Bozeman, Montana. a. How much outside surface is there to paint (not counting the bottom)? b. Express the answer to the nearest square foot.

Answer 1

The outside area of surface that can be painted is 6075π =19075.5 square feet.

a)The equation of the curve of dome  can be written as

$x^2+y^2=45^2\\\\x=\sqrt{45^2-y^2}....i$

The dome is revolving around the y-axis,

the surface area of dome about y-axis can be written as

$S=\int\limits^a_b 2\pix\sqrt{1+(\frac{dx}{dy})^2} \, dy$

The surface area of the dome shown in the figure is the sum of the area of the hemisphere 45 ft and the surface area of the band of radius 45 ft and height 22.5 ft.

Differentiating eq. i with respect to x

$\frac{dx}{dy}=\frac{1}{2}(45^2-y^2)^{\frac{1}{2}-1}(-2y)\\\\\frac{dx}{dy}=-\frac{y}{\sqrt{45^2-y^2}}$

substituting  $\frac{dx}{dy}$ in S,

and also substituting a=45 and b=-22.5

$S=\int\limits^{45}_{-22.5} \ 2\pi (\sqrt{45^2-y^2})\sqrt{1+(-\frac{y}{\sqrt{45^2-y^2}})^2} \, dy\\\\S=2\pi\int\limits^{45}_{-22.5} \sqrt{45^2-y^2} \sqrt{\frac{45^2-y^2+y^2}{45^2-y^2}}dy\\\\S=2\pi\int\limits^{45}_{-22.5}\sqrt{45^2}dy\\\\S=2\pi\ (45)(45-(-22.5))\\\\S=6075 \pi\ square\ feet$

Thus, the outside area of surface that can be painted is 6075π square feet

b)The  outside surface area is 6075π =19075.5 square feet

To learn more about Integrations refer here

https://brainly.com/question/14279102

#SPJ4