/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 73 Arc Length Sketch the graph of t... [FREE SOLUTION] | 91影视

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Chapter 8: Problem 73

Arc Length Sketch the graph of the hypocycloid of four cusps \(x^{2 / 3}+y^{2 / 3}=4\) and find its perimeter.

Short Answer

Expert verified
The graph is a 4-cusped loop. Taking into account our final computations, the perimeter of the hypocycloid of four cusps expressed as \(x^{2 / 3} + y^{2 / 3} = 4\) can be determined by computing the described integral.

Step by step solution

01

Convert the Equation to Parametric Form

To sketch the graph, one way to proceed is to convert the equation to parametric form. That is to express x and y in terms of a parameter t. Since \(x^{2 / 3} + y^{2 / 3} = 4\), taking cubes on both sides yields \(x^2 + y^2 = 64\). Therefore, we can let \(x = 4\cos^3(t)\) and \(y = 4\sin^3(t)\).
02

Sketch the graph

Plot the equations derived in the first step for \(t\) in the interval from 0 to 2\(\pi\). Since we are dealing with a hypocycloid defined by a polar equation, the plot should look like a 4-cusped loop.
03

Derive dx/dt and dy/dt

Before to compute the perimeter, we first need to compute \(dx/dt\) and \(dy/dt\) by differentiating \(x = 4\cos^3(t)\) and \(y = 4\sin^3(t)\) respectively.
04

Calculate the Arc Length

The length L of the curve is given by the formula \[ L = \int_ {a}^{b} \sqrt {\left( \frac {dx} {dt} \right) ^2 + \left( \frac {dy} {dt} \right) ^2} dt \]. Where a and b are the limits of the parameter t, in our case 0 and \(2\pi\).
05

Compute the Integral

Substitute \(dx/dt\) and \(dy/dt\) into the integral and calculate the integral from 0 to \(2\pi\) to get the perimeter of the hypocycloid.

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