Difference between revisions of "009A Sample Final 1, Problem 5"

Revision as of 09:22, 27 February 2017

A kite 30 (meters) above the ground moves horizontally at a speed of 6 (m/s). At what rate is the length of the string increasing

when 50 (meters) of the string has been let out?

Foundations:
The Pythagorean Theorem
For a right triangle with side lengths $a,b,c$ where $c$ is the length of the
hypotenuse, we have $a^{2}+b^{2}=c^{2}.$

Solution:

Step 1:
Insert diagram.
From the diagram, we have $30^{2}+h^{2}=s^{2}$ by the Pythagorean Theorem.
Taking derivatives, we get
$2hh'=2ss'.$

Final Answer:
$s'={\frac {24}{5}}$ m/s

@@ Line 8: / Line 8: @@
 |'''The Pythagorean Theorem'''
 |-
-|&nbsp; &nbsp; &nbsp; &nbsp; For a right triangle with side lengths <math style="vertical-align: -4px">a,b,c</math>, where <math style="vertical-align: 0px">c</math> is the length of the
+|&nbsp; &nbsp; &nbsp; &nbsp; For a right triangle with side lengths &nbsp;<math style="vertical-align: -4px">a,b,c</math>&nbsp; where &nbsp;<math style="vertical-align: 0px">c</math>&nbsp; is the length of the
 |-
 |
-&nbsp; &nbsp; &nbsp; &nbsp; hypotenuse, we have <math style="vertical-align: -2px">a^2+b^2=c^2.</math>
+&nbsp; &nbsp; &nbsp; &nbsp; hypotenuse, we have &nbsp;<math style="vertical-align: -2px">a^2+b^2=c^2.</math>
 |}
@@ Line 22: / Line 22: @@
 |Insert diagram.
 |-
-|From the diagram, we have <math style="vertical-align: -3px">30^2+h^2=s^2</math> by the Pythagorean Theorem.
+|From the diagram, we have &nbsp;<math style="vertical-align: -3px">30^2+h^2=s^2</math>&nbsp; by the Pythagorean Theorem.
 |-
 |Taking derivatives, we get
@@ Line 33: / Line 33: @@
 !Step 2: &nbsp;
 |-
-|If &nbsp; <math style="vertical-align: -4px">s=50,</math> then&thinsp; <math style="vertical-align: -2px">h=\sqrt{50^2-30^2}=40.</math>
+|If &nbsp; <math style="vertical-align: -4px">s=50,</math>&nbsp; then &nbsp;<math style="vertical-align: -2px">h=\sqrt{50^2-30^2}=40.</math>
 |-
 |So, we have &nbsp; <math style="vertical-align: -5px">2(40)6=2(50)s'.</math>