Nothing Special   »   [go: up one dir, main page]
Scribd Logo
Upload

Welcome to Scribd!

  • 669 views

    Lec 18 - Related Rates With Trig

    Uploaded by

    Ranniv Alexandria Ramos Salgado
    1. The document contains 7 applied mathematics word problems involving related rates and trigonometric functions. The problems involve quantities changing with respect to time such as angles, distances, heights, and speeds. The goal is to determine the rate of change of some quantity using trigonometric relationships and related rates equations.

    Copyright:

    © All Rights Reserved
    Download as PPTX, PDF, TXT or read online from Scribd

    Lec 18 - Related Rates With Trig

    Uploaded by

    Ranniv Alexandria Ramos Salgado
    669 views7 pages
    1. The document contains 7 applied mathematics word problems involving related rates and trigonometric functions. The problems involve quantities changing with respect to time such as angles, distances, heights, and speeds. The goal is to determine the rate of change of some quantity using trigonometric relationships and related rates equations.

    Original Description:

    Original Title

    lec 18 -related rates with trig.pptx

    Copyright

    © © All Rights Reserved

    Available Formats

    PPTX, PDF, TXT or read online from Scribd

    Did you find this document useful?

    Is this content inappropriate?

    1. The document contains 7 applied mathematics word problems involving related rates and trigonometric functions. The problems involve quantities changing with respect to time such as angles, distances, heights, and speeds. The goal is to determine the rate of change of some quantity using trigonometric relationships and related rates equations.

    Copyright:

    © All Rights Reserved
    Download as PPTX, PDF, TXT or read online from Scribd
    Download as pptx, pdf, or txt
    669 views7 pages

    Lec 18 - Related Rates With Trig

    Uploaded by

    Ranniv Alexandria Ramos Salgado
    1. The document contains 7 applied mathematics word problems involving related rates and trigonometric functions. The problems involve quantities changing with respect to time such as angles, distances, heights, and speeds. The goal is to determine the rate of change of some quantity using trigonometric relationships and related rates equations.

    Copyright:

    © All Rights Reserved
    Download as PPTX, PDF, TXT or read online from Scribd
    Download as pptx, pdf, or txt
    of 7

    related rates with

    trigonometric
    functions
    A man in a motorboat at A receives a message at noon, calling him to B. A
    bus making 40 miles per hour leaves C, bound for B, at 1:00 PM. If AC = 30
    miles, what must be the speed of the boat, to enable the man to catch the
    bus?
    30 sec  30 tan 
     1
    r 40
    120 sec 
    r
    3 tan   4
     
    dr 3 tan   4120 sec  tan    120 sec  3 sec 2 
     0
    d 3 tan   4 2

    Let r = rate of boat tan  3 tan   4  3 sec 2   0


    time boat travelled = time bus travelled +1 3
    tan  
    s x 4
     1 5
    r 40 120 
    r  4   24mph
    s  30 sec 3
    3   4
    4
    x  30 tan 
    Two sides of a triangle have lengths 15m and 20m. The angle between
    them is increasing at π/90 rad/s. How fast is the length of the third side
    changing when the angle between the sides is π/3?
    d 
     rad / s
    dt 90
    dx
    ? 
    
    dt 3
    using cosine law: c 2  a 2  b 2  2ab cos C
    x 2  15  20  21520 cos 
    2 2

    x 2  625  600 cos 


    differentiating with respect to t:
    dx  d 
    2x  600  sin  
    dt  dt 
    dx  d  at θ = π/3
    2x  600  sin  
    dt  dt  
    x 2  625  600 cos
    3


    2 5 13
    dx
    dt
       
     (600 sin ) 
    3  90 
    x  5 13

    dx
     0 .5 m / s
    dt
    As the sun rises, the shadow cast by a 15 m tree is decreasing at a rate of 55
    cm/h. At what rate is the angle of elevation from the shadow to the sun
    increasing when the shadow is 15 m in length?

    dx  1m 
     55cm / h   0.55m / h
    dt  100cm 
    d
    ? when x = 15 m
    dt 
    15
    tan  
    15
    
    tan   15 4
    x
     d   2  dx 
    sec  
    2
      15 x  
     dt   dt 
      d
    2

     1515  0.55
    2
     sec 
     4  dt
    d 11
     rad / h
    dt 600
    1. The base of a right triangle grows 2 ft/sec, the altitude grows 4 ft/sec. If
    the base and altitude are originally 10 ft and 6 ft, respectively, find the time
    rate of change of the base angle, when the angle is 45°.

    2. A rowboat is pushed off from a beach at 8 ft/sec. A man on shore holds


    a rope, tied to the boat, at a height of 4 ft. Find how fast the angle of
    elevation of the rope is decreasing, after 1 sec.

    3. A kite is 60 ft high with 100 ft of cord out. If the kite is moving


    horizontally 4 mi/hr directly away from the boy flying it, find the rate of
    change of the angle of elevation of the cord.

    4. A ship, moving 10 mi/hr, sails east for 2 hours, then turns N 30° E. A
    searchlight, placed at the starting point, follows the ship. Find how fast
    the light is rotating (a) 4 hours after the start; (b) just after the turn.
    5. An observer watches a rocket launch from a distance of 2 km . The angle
    of elevation, θ , is increasing at 3º per second at the instant when
    θ = 45º. How fast is the rocket clmbing at that instant?
    6. The beacon of a lighthouse 1 km from a straight shore revolves 5 times
    per minute and shines a spot of light on the shore. How fast is the spot of
    light moving when θ = 30º.

    7. A train is travelling at 4/5 kmper min along a straight track, moving in the
    direction shown in the figure above. A movie camera positioned 1 km from
    the track is focused on the train. ow fast is the distance z from the camera to
    the train changing when the train is 2 km from the camera? b. When the
    train is 2 km from the camera θ = π/3. How fast is the camera rotating at
    the moment when the train is 2 km from the camera?

    You might also like