HW07-solutions

# HW07-solutions - zakaria (mmz255) HW07 gilbert (55485) 1...

This preview shows pages 1–4. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: zakaria (mmz255) HW07 gilbert (55485) 1 This print-out should have 20 questions. Multiple-choice questions may continue on the next column or page find all choices before answering. 001 10.0 points Find a Cartesian equation for the curve given in parametric form by x ( t ) = 8 t 2 , y ( t ) = 8 t 3 . 1. x = 4 y 2 / 3 2. x = 2 y 3 / 2 3. x = 2 y 4 / 3 4. x = 2 y 2 / 3 correct 5. x = 4 y 4 / 3 6. x = 4 y 3 / 2 Explanation: We have to eliminate the parameter t from the equations for x and y . But from the equation for y , it follows that t = 1 2 y 1 / 3 , in which case x = 8 parenleftbigg 1 2 y 1 / 3 parenrightbigg 2 = 2 y 2 / 3 . 002 10.0 points Describe the motion of a particle with posi- tion P ( x, y ) when x = 5 sin t , y = 4 cos t as t varies in the interval 0 t 2 . 1. Moves along the line x 5 + y 4 = 1 , starting at (0 , 4) and ending at (5 , 0). 2. Moves once clockwise along the ellipse x 2 25 + y 2 16 = 1 , starting and ending at (0 , 4). correct 3. Moves once clockwise along the ellipse (5 x ) 2 + (4 y ) 2 = 1 , starting and ending at (0 , 4). 4. Moves along the line x 5 + y 4 = 1 , starting at (5 , 0) and ending at (0 , 4). 5. Moves once counterclockwise along the ellipse x 2 25 + y 2 16 = 1 , starting and ending at (0 , 4). 6. Moves once counterclockwise along the ellipse (5 x ) 2 + (4 y ) 2 = 1 , starting and ending at (0 , 4). Explanation: Since cos 2 t + sin 2 t = 1 for all t , the particle travels along the curve given in Cartesian form by x 2 25 + y 2 16 = 1 ; this is an ellipse centered at the origin. At t = 0, the particle is at (5 sin0 , 4 cos0), i.e. , at the point (0 , 4) on the ellipse. Now as t increases from t = 0 to t = / 2, x ( t ) increases from x = 0 to x = 5, while y ( t ) decreases from y = 4 to y = 0 ; in particular, the particle moves from a point on the positive y-axis to a point on the positive x-axis, so it is moving clockwise . In the same way, we see that as t increases from / 2 to , the particle moves to a point zakaria (mmz255) HW07 gilbert (55485) 2 on the negative y-axis, then to a point on the negative x-axis as t increases from to 3 / 2, until finally it returns to its starting point on the positive y-axis as t increases from 3 / 2 to 2 . Consequently, the particle moves clockwise once around the ellipse x 2 25 + y 2 16 = 1 , starting and ending at (0 , 4). keywords: motion on curve, ellipse 003 10.0 points Which one of the following could be the graph of the curve given parametrically by x ( t ) = t 3 2 t , y ( t ) = t 2 3 , where the arrows indicate the direction of increasing t ? 1. x y correct 2. x y 3. x y 4. x y 5. x y 6. x y Explanation: All the graphs are symmetric either about the y-axis or the x-axis. Lets check which it is for the graph of ( x ( t ) , y ( t )) = ( t 3 2 t, t 2 3) . zakaria (mmz255) HW07 gilbert (55485) 3 Now x ( t ) = ( t ) 3 2( t ) = ( t 3 2 t ) = x ( t ) , and y (...
View Full Document

## This note was uploaded on 06/05/2011 for the course MATH 408 D taught by Professor Gilbert during the Spring '11 term at University of Texas at Austin.

### Page1 / 13

HW07-solutions - zakaria (mmz255) HW07 gilbert (55485) 1...

This preview shows document pages 1 - 4. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online