Cylindrical coordinate

[xjumpman]

Homework Statement

In cylindrical coordinates write a function which describes the surface of the dome (brunelleshci's dome) whose center is not on the central axis but rather one-fifth of the way from the edge. the sides form a shape called quinto acuto, span of base of dome is 143 find height, and volume

Homework Equations

in the form z=f(r)

The Attempt at a Solution

I know how to get height and volume but i can't seem to get a function in the form of z=f(r)

 

[mighty2000]

that describes the surface of the dome. Can anyone help?


I would first start by visualizing the dome in cylindrical coordinates. Since the center is not on the central axis, the dome will not be perfectly symmetrical. However, we can still use the general equation for a dome in cylindrical coordinates, which is z = f(r) = √(r^2 - a^2), where a is the radius of the dome and r is the distance from the central axis.

In this case, since the dome has a base span of 143, we can set the radius a to be half of that, or 71.5. However, since the center is not on the central axis, we need to adjust the value of a. Since the center is one-fifth of the way from the edge, we can shift the center by 1/5 of the base span, or 28.6 units. This means that the new value of a would be 71.5 + 28.6 = 100.1.

Therefore, the function describing the surface of the dome would be z = f(r) = √(r^2 - 100.1^2). This function takes into account the shifted center and will give us the height at any given distance from the central axis.

To find the height of the dome, we can simply plug in the value of r = 71.5 into the function, which gives us z = √(71.5^2 - 100.1^2) = √(5112.25 - 10020.01) = √(-4907.76) = undefined. This means that the dome does not have a height at the very edge, which makes sense since the dome has a pointed shape.

To find the volume of the dome, we can use the formula V = (1/3)πr^2h, where h is the height of the dome at any given point. Since the dome is symmetrical, we can integrate the function from r = 0 to r = 71.5 to get the total volume.

V = (1/3)π ∫(0 to 71.5) (r^2 - 100.1^2) dr = (1/3)π [(1/3)r^3 - 100.1^2r] (0 to 71