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1) A parabola with equation y=x^2+bx+c passes through the points (-1,-11) and (3,17). What is c?

 

2)A point P is randomly placed in the interior of the right triangle below. What is the probability that the area of triangle PBC is less than half of the area of triangle ABC? Express your answer as a common fraction,

3) The first six rows of Pascal's triangle are shown below, beginning with row zero. Except for the 1 at each end, row 4 consists of only even numbers, as does row 2. How many of the first 20 rows have this property? (Don't include row 0 or row 1).

 

 

 

THX FOR THE HELP IN ADVANCE

 Aug 15, 2019
edited by Guest  Aug 15, 2019
 #1
avatar+102913 
+1

1) A parabola with equation y=x^2+bx+c passes through the points (-1,-11) and (3,17). What is c?

 

We have that

 

(-1)^2  + b(-1)  + c  = -11  ⇒   1 -b + c  = -11  ⇒   -b + c  = -12       (1)

and

(3)^2   + b(3) + c  = 17  ⇒  9 + 3b + c  = 17  ⇒  3b + c  = 8  ⇒  -3b - c  = -8    (2)

 

Add (1)  and (2)  and we get that

 

-4b  = -20

b = 5

 

So....using (1)     -b + c   =-12

 

-5 + c  = -12     subtract 10 from both sides

 

c  =   -7

 

 

cool cool cool

 Aug 15, 2019
 #2
avatar+102913 
+1

The Area of triangle ABC   = (1/2)(B)(AB)

 

Draw  DE  parallel to CB  such that DE is  at a height of (1/2)AB

 

We have a trapezoid DEBC....the area of this trapezoid is  

 

(1/2) (1/2) (AB)(( DE + BC)

But using similar triangles....DE   =  1/2BC

 

So...the area of  trapezoid DEBC  =  (1/2) (1/2)(AB) ( [1/2 BC] + BC)  =  (1/4)(AB)( 3/2 )(BC)  = (3/8)(AB)(BC)

 

And if  P is located inside this trapezoid.....the area of triangle PBC will be less than (1/2) that of  ABC

 

So.....the  probability  is   [ DEBC] / [ABC ]  =  (3/8)(AB)(BC) /  [ (1/2) (AB) (BC) ] =  (3/8)/(1/2)  =  3/4

 

 

cool cool cool

 Aug 15, 2019
edited by CPhill  Aug 15, 2019
edited by CPhill  Aug 15, 2019
edited by CPhill  Aug 15, 2019
 #3
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Thx for the help but it says that the answer is incorrect for question \(2\). The answer is not \(\frac{1}{2}\)Srry. 

Guest Aug 15, 2019
 #6
avatar+102913 
0

Oops...see my corrected answer  !!!

 

 

cool cool cool

CPhill  Aug 15, 2019
 #7
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THANK YOU SO MUCH!laugh

Guest Aug 15, 2019
 #4
avatar+102913 
+1

3) The first six rows of Pascal's triangle are shown below, beginning with row zero. Except for the 1 at each end, row 4 consists of only even numbers, as does row 2. How many of the first 20 rows have this property? (Don't include row 0 or row 1).

 

The even rows  occur at  the  2^n  rows    where  n is an integer  ≥ 1

 

So    rows   2, 4, 8, 16    have the property  that all the entries except fo the  beginning and ending 1s  are even

 

 

cool cool cool

 Aug 15, 2019
 #5
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0

THX FOR ALL OF THE HELP. These questions had me stuck for a long time. THX

 Aug 15, 2019

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