Application of square function in real life

APPLICATIONS OF THE QUADRATIC FUNCTION IN REAL LIFE APLICAȚII ALE FUNCTIEI PĂTRATICE IN VIAȚA COTIDIANĂ

Jean Monnet High School Bucharet Grade 9C eTwinning Project 2017-2018

ANGRY BIRDS GAMES If you have an iphone you should probably know that the height of the display is 4.5 inches and the width 2.31 inches. So if you are playing Angry Birds you can make a function in order to pass every level. You can see an example below .

We know that the birds has the coordinates (0,0) , the point where we want to hit in this example is the top of the function (2.86; 0.57) and the place were the bird should fall if it doesn’t hit anything would be (5.71; 0).

Find f, f ( x )=a x 2 +bx +c C(0; 0):

⇒ f ( 0 )=0 ⇒ c=0 {x=0 y=0

−b ⟹ x v =2,86 ⟹ b=−5,72 a 2a −∆ −b2 −4 ac y v= ⟹ y v= =0,57⟹ a ≅−0,07⟹ b=−5,72∙(−0,07)=0,40 4a 4a

x v=

f(x) = –0.07 x² + 0.4 x

A HIGHWAY UNDERPASS A highway underpass is parabolic in shape. If the curve of the underpass can be modeled by h:R→R, h (x) = 50 – 0.02x2 , where x and h(x) are in feet.

How high is the highest point of the underpass, and how wide is it?

O trecere subterană este parabolică. Dacă curba trecerii poate fi modelată de funcția: h:R→R, h (x) = 50 – 0.02x2, unde x și h (x) sunt exprimate în picioare.

Cât de mare este punctul cel mai înalt al podului și cât de lat este?

V

−∆ ; ( −b 2a 4a )

este punctul de maxim al funcției h ⇒ V ( 0 ; 50 )

G h ∩Ox ⇒ 50−0,02 x 2=0 ⇒ x 1,2=

−b ± √ ∆ ⇒ x 1,2=± 50 ⇒ A (−50 ; 0 ) , B ( 50 ; 0 ) ⇒ 2a AB=100

ARCHITECTURE Did you know that the two main cables from The Golden Gate Bridge were calculated/made with a function ?

Length of main span portion of suspended structure (distance between towers): 4200 ft = 1280 m Height of tower above roadway: 500 ft = 152,4m Find f, f ( x )=a x 2 +bx +c x=0, y=152 ⟹ f ( 0 )=152 ⟹ c=152,4 −b 2a b=−1280 a x v =640=

;

2

y v =0=

−b −4 ac 4a

12802 a2−4 a 152=0 ⟹ a=

4 ∙152 =0,00038 1280 ∙ 1280 ⟹ a=0,0004

⟹ b=−0,475

f ( x )=0,00038 x 2−0,475 x +152

SPORTS At a basketball team a man is training to throw the ball into the basket. We now that the ball in his hands is at a height of 2 meters above the ground, the basket is at 3 meters far away from the man and at a height of 3 meters. At what height and how far away from the man is going to reach the top point ?

Find f, f ( x )=a x 2 +bx +c x=0 x=3 x=1 y=2 y=3 y =6

{

c=2 9 a+3 b=1 9 a+3 b+2=3 ⟺ a+b=4 a+b +2=6

a=4−b ⇒36−9 b+ 3 b=1 ⇒ b=

−b 35 = =1,59 2 a 22 −∆ y v= =6,58 4a x v=

{

35 −11 ⇒ a= 6 6 11 35 ⇒ f ( x )= x2 + x +2 6 6

HISTORY Did you knew that the romans invented aqueducts so that they could carry water from long distances . One of this aqueducts is in France .

In order to support the water there were made parabolas so that the bridge can resist . What is the function of these parabolas?

A(0,0) B(4,0) C(2;1,4) −b =2⟹ b=−4 a 2a 2 −b −4 ac y v= =1,4 ⟹ a=−0,35⟹ b=1,4 4a

xv=

2

f ( x )=−0,35 x +1,4 x

CEL MAI BUN PREȚ/THE MOST PROFITABLE PRICE O firmă de încălțăminte de damă, tocmai înființată, vrea să afle care este cel mai profitabil preț la care să își vândă produsul. Știind următoarele lucruri, află prețul cel mai potrivit. Costul înființării firmei este de 700000 lei, fiecare pereche de pantofi costa 110lei să fie făcută și se mai știe de la firme similare că produsele s-ar vinde după acest model: 70,000–200P, P fiind prețul.

A new company selling women's shoes wants to find out what the most profitable price point to sell their product at is. Knowing the following, find out the best price.

The cost of setting up the company is 700,000 lei, each pair of shoes costs 110lei and it is also known from similar companies that the products would be sold according to this model: 70,000-200P. P being the price.

Solution: Shoe sales= 70000−200 P Sales in lei ¿ ( 70000−200 P ) × P=70000 P−200 P2 Costs= 700,000+110 × ( 70,000−200 P )=700000+7700000−22000 P ¿ 840000−22000 P Profit= sales – costs ¿ 70000 P−200 P2−8400000−22000 P −200 P2 +92000 P−8400000⟹ f ( x )=−200 x 2+ 92000 x−8400000 a=−200 b=92000 c=−8400000 2 −200 P +92000 P−8400000 , dar we divide everything by 200 ⟹ a=−1 b=460 c=−42,000 f : [ 125.6 ; 334.4 ] →[0; 10900] −b −460 = =230 lei 2a −2

NATURE In nature you can find plants that describe a parabola for example the leafs of the palm tree but also the petals of the Waterlily form a parabola.

x v =0,82=

−b ⟹ b=−1,64 a 2a 2

y v =2,5=

−b −4 ac −10 ⟹ a= =−3,73 ⟹b=6,11 4a 2,68

f ( x )=−3,73 x2 +6,11 x

Equation for square function

A (−1 ; 1,5 ) ∈Gf ⇒ f (−1 )=1,5 B ( 0 ; 2 ) ∈G f ⇒ f ( 0 ) =2 C ( 1 ; 1 ) ∈G f ⇒ f ( 1 ) =1

{

a∙(−1)2 +b∙ (−1 ) + c=1,5 c=2 a∙ 1+ b+c =1

{a−b+2=1,5 a+b +2=1

{

⇒

{

a−b +c=1,5 c=2 a+b+ c=1 ⇒b=1−c−a

⇒ a−b=−0,5 ⇒2 a=−1,5 ⇒ a=−0,75 a+ b=−1 b=1−c−a=1−c +0,75=1+ 0,75=−0,25 c=2 ⇒ f ( x )=−0,75 x 2−0,25 x+ 2