Scene 1 (0s)
LINEAR PROGRAMMING. BY:SUZANNA NELSON.
Scene 2 (7s)
OBJECTIVEs. Identify the variables and the constraints. Find the objective function Graph the solution Find the maximum profit.
Scene 3 (20s)
Solve the word problem by creating and solving a system of linear equations.
Scene 4 (1m 54s)
ANSWER. METHOD 1 BY ELIMINATION METHOD Orange daze smoothie contains Juice orange=10 fluid ounces + Juice pineapple= 4 fluid ounces + Juice blueberry=2 fluid ounces.
Scene 5 (2m 12s)
ANSWER CONTINUED. (ii)Pineapple blue smoothie contains Juice orange=5 fluid ounces + Juice pineapple= 6 fluid ounces + Juice blueberry=4 fluid ounces.
Scene 6 (2m 30s)
ANSWER CONTINUED. Now let, Orange daze=x and pineapple blue=y According to data given we will set up systems of equations 10x +5y=500 equation 1 4x+6y=360 equation 2 2x+4y=250 equation 3.
Scene 7 (2m 37s)
ANSWER CONTINUED. Now taking equation 1 and 2 To make both equations equal by multiplying the first equation by 4 and the second equation by 10 10x+5y=500 *4 4x+6y=360*10 Simplify 40x+20y=2000 40x+60y=3600 Subtract 40x+60y=3600 from 40x+20y=2000 by subtracting like terms on each side of the equal sign. 40x−40x+20y−60y=2000−3600 Add 40x to − 40x so the x terms can cancel out 20y−60y=2000-3600.
Scene 8 (4m 30s)
ANSWER CONTINUED. Add 20y to −60y. −40y=2000−3600 Add 2000 to −3600. −40y=−1600 Divide both sides by −40 . -40y/-40=1600/-40 y=40.
Scene 9 (4m 44s)
ANSWER CONTINUED. Substitute y=40 to solve x in equation 2 4x+6y=360 4x+6×40=360 4x+240=360 Subtract 240 from both sides of the equation. 4x=120 Divide both sides by 4 . 4x/4=120/4 x=30 By Elimination method x=30 and y=40.
Scene 10 (5m 0s)
Taking equation 2 and 3 4x+6y=360 →(1) 2x+4y=250 →(2 ) E liminate the variable x. 4x+6y=360×1→4x+6y=360 − 2x+4y=250×2→4x+8y=500 -2y=-140→ (3).
Scene 11 (5m 12s)
ANSWER CONTINUED. Using substitution method From (3) -2y=-140 ⇒ y =-140/-2 = 70 From (2 ) 2x+4y=250 ⇒2x+4(70)= 250 ⇒ 2x+280=250 ⇒2x=250-280=- 30 ⇒x =-30/2 = -15 Solution using Elimination method x =- 15 and y=70.
Scene 12 (5m 28s)
ANSWER CONTINUED. Taking equation 1 and 3 Eliminate the variable x. 10x+5y=500×1→10x+5y =500 − 2x+4y=250×5 → 10x+20y=1250 -15y=-750→ (3).
Scene 13 (5m 39s)
ANSWERED CONTINUED. Using substitution method From (3) -15y=- 750 ⇒y =-750/-15=50 From (2) 2x+4y=250 ⇒2x+4(50)= 250 ⇒ 2x+200=250 ⇒ 2x=250-200=50 ⇒ x=50/2 = 25 By Elimination method x=25 and y=50..
Scene 14 (5m 54s)
ANSWER CONTINUED. Since the store makes a profit of $1.50 on each bottle of orange daze and $1 on each bottle of pineapple blue $1.50*25=$37.50 $1*50=$50 $37.50+$50=$87.50 Hence, the maximum profit is $87.50 when the store makes 25bottles of orange daze and 50 bottles of pineapple blue..
Scene 15 (7m 39s)
ANSWER CONTINUED. desmos Graph 80 (o, 62.5) ICh- + 5y 500 (-15, 70) @ •LX+6y=360 (25, 50) 2x+ 4y — 250 40 (30, 40) -20 desmos -20 100.
Scene 16 (8m 24s)
ANSWER CONTINUED. METHOD 2.
Scene 17 (8m 32s)
ANSWER CONTINUED. Objective function Profit :P=1.50x+1y Constraint/Inequality/Subject to: x,y ≤0 10x+5y ≤500 4x+6y ≤360 2x+4y ≤ 250.
Scene 18 (8m 45s)
ANSWER CONTINUED. Sketch the Graph. Untitlexj Graph ION + 500 + O, 360 250 desmos desmos (-15, 70) (o. 60) .20 09 .40 (25.50) (30, 40) 20.
Scene 19 (9m 0s)
ANSWER CONTINUED. Points on graph A(-15,70) ,B(0,60) ,C(30,40) and D(25,50) P(A)=1.50(-15)+1(70)=$47.50 P(B)=1.50(0)+1(60)=$60.00 P(C)=1.50(30)+1(40)=$85.00 P(D)=1.50(25)+1(50)=$87.50 Profit P(D)= 1.50(25)+1(50)=$87.50 Hence,25 x and 50 y should be used to make the maximum profit..
Scene 20 (9m 28s)
Applications OF LINEAR PROGRAMMING TO REAL LIFE SITUATIONS.
Scene 21 (9m 33s)
PERSONAL REFLECTION. Linear programming has proven to be a powerful tool widely used to solve optimization ,both in modeling real-word problems and has a widely applicable mathematical theory. Linear programming apply mathematical models to linear problems in order to maximize or minimize an objective function respecting some constraints. To do this we can assign constraints to the functions and use to it find the maximum or minimum profit. This unit has thought me decision making to make quality decisions that will make use of the optimum resources and allow the company to make profits. It has also thought me to apply the techniques learnt to solve practical problems..