Section 2.7 Related Rates (Word Problems)

1 Section Related Rates (Word Problems) The idea is to compute the rate of change of one quantity in terms of the rate of change of anotherquantity.(pg. 127) Example: Air is being pumped into a spherical balloon so that its volume increases at a rateof 100 cm3 / s. How fast is the radius of the balloon increasing when the diameter is 50 cm?Step 1 & 2: Draw a picture and list what you (units tell variables)d=2r=50cmr=25cmFind: drdtStep 3: Write an equation relating the r3 Step 4: Take derivatives of both sides (implicitly).ddt V =ddt 43 r3 dVdt=43 3r2 drdt=4 r2drdtStep 5 Solve for what you are trying to r2 dVdtStep 6 Substitute in what you 25 2 100=25 25 2=125 Step 7: Check the cms125 cmsNote: There are 5 very nice examples worked out in the textbook.

2 (pg. 131, #4) Example: The length of a rectangle is increasing at a rate of 8 cm / s and its width is increasing at a rate of 3 cm / s. When the length is 20 cm and the width is 10 cm, how fast is the area of the rectangle increasing?1. Draw Picture:2. List what you know:dldt=8cmsdwdt=3cms3. Write an equation:A=l w4. Take derivatives:ddt A =ddt lw dAdt=ldwdt wdldt5. Solve for what you need: Already in the form we Substitute in:dAdt=ldwdt wdldtdAdt=20cm 3cms 10cm 8cmsdAdt=60cm2s 80cm2sdAdt=140cm2s7. Check units:dAdt cm2sall : A street light is mounted at the top of a 15 ft tall pole.

3 A man 6 ft tall walks away from the pole with a speed of 5 ft / sec along a straight path. How fast is the tip of his shadowmoving when he is 40 ft from the pole?1. Draw Picture:2. List what you know:dxdt=5ftswhen x = 40 ft3. Write an equation:Notice Similar Triangles ABE and :156=x yy15y=6 x y 15y=6x 6y9y=6xy=23xAside: y=23 40 =8034. Take derivatives: (tough part)Key: The tip of the shadow moves at a rate of ddt x y ddt x 23x =ddt 53x =53dxdt5. Solve for what you need: Already in the form we need6. Substitute in:ddt x y =53 5fts =253fts7. Check units:ddt x y =ftsOK.