math1a - quiz 9 - solns

# math1a - quiz 9 - solns - Math 1A Quiz 9 Solutions April...

This preview shows pages 1–2. Sign up to view the full content.

Math 1A Quiz 9 Solutions April 21st, 2008 1. Find the area of the largest rectangle with sides parallel to the coor- dinate axes which can be inscribed in the ellipse x 2 + y 2 4 = 1. Solution: Let ( x, y ) be the coordinates of the corner of the rectangle which lies in the ﬁrst quadrant. Then the area of the rectangle is A = 4 xy . We want to maximize A with respect to the constraint x 2 + y 2 4 = 1. Solving this constraint for y gives y = 4 - 4 x 2 = 2 1 - x 2 (Note we don’t have to worry about signs here since we only care about positive x and y ). So we get the formula A = 4 xy = 8 x 1 - x 2 . Now, taking derivatives gives: A 0 ( x ) = 8 p 1 - x 2 + 8 x 1 2 · 1 1 - x 2 · ( - 2 x ) = 8 p 1 - x 2 - 8 x 2 1 - x 2 = 8 - 16 x 2 1 - x 2 . The critical points of A ( x ) occur either when A 0 ( x ) = 0 or A 0 ( x ) does not exist. These correspond to the roots of the numerator and denominator of A 0 , respectively. The roots of the numerator are at x = ± 2 2 (we only care about the positive solution) and

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.

{[ snackBarMessage ]}

### Page1 / 2

math1a - quiz 9 - solns - Math 1A Quiz 9 Solutions April...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online