Lesson 16 MA 152, Section 2.5

Lesson 16 MA 152, Section 2.5

<p> Lesson 16 MA 152, Section 2.5</p><p>The quotient of two numbers or quantities is called a ratio. Ratios often are used for comparisons. For example, if there are 120 pieces of candy for 13 students, that would be a ratio of 120 . 13</p><p>An equation indicating two ratios are equal is called a proportion. In any proportion, cross products are equal. a c  b d  a   c  (bd)   (bd)   b   d  ad  bc</p><p>Ex 1: Solve each problem by writing and solving a proportion. At a fraternity gathering, there were 2 males for every 3 females. If there were 24 females, how many were males?</p><p>Two variables are said to vary directly or be directly proportional if their ratio is constant. For example: examine the following table where number of hours worked is compared with earnings. # of hours earnings 1 $13 3 $39 8 $104 20 $260 260 104 39 13    These ratios are equal. 20 8 3 1 We say that the earnings vary directly as the number of hours worked. y  k  If y varies directly as x: x where k is some constant. y  kx</p><p>For k > 0: In direct variation, as x increases so does y; or, as x decreases, so does y.</p><p>1 Definition: Direct Variation: The words 'y varies directly with x' or 'y is directly proportional to x' leads to the equation y  kx , for some constant value k. The number k is called the constant of proportionality or the variation constant.</p><p>Two variables are said to vary inversely or be inversely proportional if their product is a constant. Examine this table where time of a trip is compared with bus speed. bus speed time of trip 20 mph 1 hr. 40 mph ½ hr. 1 60 mph hr. 3 80 mph ¼ hr.</p><p>(80)(1 )  (60)(1)  (40)( 1 )  (20)(1) The products are equal. 4 3 2 We say that the bus speed varies inversely as the time of the trip.</p><p> xy  k  If y varies inversely as x: k where k is some constant. y  x</p><p>For k > 0: In inverse variation, as x increases (for x > 0), y decreases; or, as x decreases ( for x > 0), y increases.</p><p>Definition: Inverse Variation: The words 'y varies inversely as x' or 'y is inversely proportional to x' k means that y  for some constant k. Again, k is called the constant of proportionality x or variation constant.</p><p>Ex 2: Find the constant of proportionality for each. Write the resulting variation equation. a) y is directly proportional to x. If x = 30, then y = 15.</p><p> b) R varies inversely as the square of I. If I = 25, then R = 100.</p><p>2 Definitions:</p><p>Joint Variation: The words 'y varies jointly with w and x' means that y = kwx for some constant k. Combined Variation: The words 'y varies directly as w and inversely as x' means that kw y  for a constant k. x</p><p>To solve a variation problem: k 1. Translate the problem into a variation format ( y  kx, y  , y  kwx, etc. ) x 2. Replace the given numbers and solve for the variation constant. 3. Re-write the variation format replacing the value of k. This is the variation equation. 4. Use that equation to solve the problem.</p><p>Solve each problem. 7 Ex 3: y is directly proportional to x. If y = 15 when x = 4, find y when x = . 5</p><p>Ex 4: q varies jointly with the square of m and the square root of n. If q = 45 when m = 3 and n = 25, find q when m = 4 and n = 36.</p><p>Ex 5: The time to drive a certain distance (t) varies inversely to the rate of speed (r). Mary drives 47 miles per hour for 4 hours. How long would it take her to make the same trip at 55 mph?</p><p>3 Ex 6: The intensity of illumination on a surface varies inversely as the square of the distance from the light source. A surface is 12 meters from a light source and has an intensity of 2. How far must the surface be from the light source to receive twice as much intensity of illumination?</p><p>Ex 7: Hooke's law states that the force required to stretch a spring a distance is directly proportional to the distance. A force of 3 newtons stretches a spring 15 centimeters. A force of 4 newtons would stretch the spring how far?</p><p>Ex 8: The power, in watts, dissipated as heat in a resistor varies directly with the square of the voltage and inversely with the resistance. If 20 volts are placed across a 20-ohm resistor, it will dissipate 20 watts. What voltage across a 10-ohm resistor will dissipate 40 watts?</p><p>4</p>

View Full Text

Details

  • File Type
    pdf
  • Upload Time
    -
  • Content Languages
    English
  • Upload User
    Anonymous/Not logged-in
  • File Pages
    4 Page
  • File Size
    -

Download

Channel Download Status
Express Download Enable

Copyright

We respect the copyrights and intellectual property rights of all users. All uploaded documents are either original works of the uploader or authorized works of the rightful owners.

  • Not to be reproduced or distributed without explicit permission.
  • Not used for commercial purposes outside of approved use cases.
  • Not used to infringe on the rights of the original creators.
  • If you believe any content infringes your copyright, please contact us immediately.

Support

For help with questions, suggestions, or problems, please contact us