Astroid Problem: The star-shaped curve in Figure 4-7h is called an astroid. Its

Chapter 4, Problem 6

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Astroid Problem: The star-shaped curve in Figure 4-7h is called an astroid. Its parametricequations arex = 8 cos3 ty = 8 sin3 t Figure 4-7ha. Confirm by grapher that these equationsgive the graph in Figure 4-7h.b. Find an equation for dy/dx.c. Evaluate the point (x, y) when t = 1, and finddy/dx when t = 1. On a copy of Figure 4-7h,draw a line at this point (x, y) that has slopedy/dx. Is the line tangent to the graph?d. At each cusp, dy/dx has the indeterminateform 0/0. Explain the difference in behaviorat the cusp at the point (8, 0) and at thecusp at the point (0, 8).e. Eliminate the parameter t. To do thistransformation, solve the two equations forthe squares of cos t and sin t in terms of xand y, then use the Pythagorean propertyfor sine and cosine.