HW2-solutions

# HW2-solutions - merino(aem2588 – HW2 – chen –(55405 1...

This preview shows pages 1–3. Sign up to view the full content.

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: merino (aem2588) – HW2 – chen – (55405) 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 A machinist has to manufacture a circular metal disk with area 930 π sq. cms. How close to the exact radius must the machinist control the radius if he is allowed an error tolerance of ± 8 π sq. cms. in the area of disk? 1. within approximately 0 . 1328 cm 2. within approximately 0 . 1288 cm 3. within approximately 0 . 1318 cm 4. within approximately 0 . 1298 cm 5. within approximately 0 . 1308 cm cor- rect Explanation: The area of a circular disk of radius r is πr 2 . When the area is 930 π , therefore, the exact radius, r ext , of the disk is r ext = √ 930 . If, however, the machinist is allowed to make the disk so that its area A is within ± 8 π of 930 π then A must satisfy the inequalities − 8 π ≤ A − 930 π ≤ 8 π . Thus the radius r the machinist makes the disk must satisfy the inequalities − 8 ≤ r 2 − 930 < 8 , in other words, √ 922 < r < √ 938 . Hence the area of the disk that the machinist makes will be within ± 8 π sq. cms. of 930 π sq.cms. when r ext − r < √ 930 − √ 922 ≈ . 1314 , and r − r ext < √ 938 − √ 930 ≈ . 1308 . Consequently, the radius r must be within approximately 0 . 1308 cms of the exact value √ 930. 002 10.0 points Find the value of b , b ≥ 0, for which lim x → braceleftBig √ 5 x + b − 5 x bracerightBig exists. 1. b = 27 2. b = 28 3. b = 26 4. b = 25 correct 5. b = 24 Explanation: We are told that lim x → braceleftBig √ 5 x + b − 5 x bracerightBig = A for some value A , but we aren’t told what the particular value of A is. The question requires us to see exactly what value b must take for A to exist. To begin, note that √ x + y − √ z = x + y − z √ x + y + √ z . merino (aem2588) – HW2 – chen – (55405) 2 Thus √ 5 x + b − 5 x = 5 x + b − 25 x ( √ 5 x + b + 5) = 5 x x ( √ 5 x + b + 5) + b − 25 x ( √ 5 x + b + 5) . Now lim x → √ 5 x + b + 5 = √ b + 5 , so by properties of limits, lim x → 5 x x ( √ 5 x + b + 5) = lim x → 5 √ 5 x + b + 5 = 5 √ b + 5 . Consequently, once again by the properties of limits, lim x → braceleftBig b − 25 x ( √ 5 x + b + 5) bracerightBig = lim x → parenleftBig √ 5 x + b − 5 x − 5 x x ( √ 5 x + b + 5) parenrightBig = A − 5 √ b + 5 . But lim x → braceleftBig b − 25 x ( √ 5 x + b + 5) bracerightBig = ( b − 25) lim x → braceleftBig 1 x ( √ 5 x + b + 5) bracerightBig . Since lim x → 1 x ( √ 5 x + b + 5) doesn’t exist, however, the only way lim x → braceleftBig b − 25 x ( √ 5 x + b + 5) bracerightBig = A − 5 √ b + 5 can hold is if b − 25 = 0 ....
View Full Document

## This note was uploaded on 02/23/2011 for the course MATH 408C taught by Professor Knopf during the Spring '10 term at University of Texas.

### Page1 / 11

HW2-solutions - merino(aem2588 – HW2 – chen –(55405 1...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online