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Can you solve these?
1.

2.

3.

 Apr 5, 2024
 #1
avatar+128794 
+1

1.

Let O be the center of the semi-circle

Draw CE  perp  to AB

Connect OC  = r

OE = 8.5

EB = r - 8.5

CB = 3

 

Using the Pythagorean Theorem

 

Height of triangle OCE   = Height of triangle ECB

 

CE^2  =  CE^2

 

r^2  - (OE)^2  = (CB)^2  - ( EB)^2

 

r^2 - (8.5)^2  =  (3)^2 - (r - 8.5)^2

 

r^2  - 72.25  =  9 - r^2 + 17r - 72.25

 

r^2 - 72.25  =  -r^2 + 17r - 63.25

 

2r^2 - 17r -9  = 0

 

(2r + 1) ( r -  9)  =  0

 

r =  9 

 

cool cool cool

 Apr 6, 2024
 #2
avatar+128794 
+1

2. 

Area of Sector  = pi *r^2 *  (100  / 360)

 

250 pi  =  pi * r^2 * (5/18)

 

r^2  = 250 (18 / 5)

 

r^2  =  50 * 18

 

r^2  = 100 * 9

 

r^2 = 900

 

r  = 30

 

cool cool cool

 Apr 6, 2024
 #3
avatar+128794 
+1

3.

 

2* area ABC =  AD * BC                  2* area ABC =  BE * AC

2* area ABC =  12 * BC                   2* area ABC  = 16 * AC

area ABC = 6 BC                             area ABC  =  8 AC

 

6 BC =  8 AC

 

AC = (8/6) BC = (4/3) BC

 

Triangle inequality

 

AB + BC >  AC

 

AB + BC >  (4/3)BC

 

AB  >  (4/3)BC - BC

 

AB > (1/3)BC

 

Then

 

2 [ ABC ]  =   BC  *  AD

 

2[ ABC ]   = BC * 12

 

2 [ ABC ] = (1/3)BC *  36

 

Since AB > (1/3)BC

 

2[ ABC ]  < AB *  36

 

2[ ABC ] < AB * CF

 

Then CF  < 36

 

So

 

Max integer value for CF =  35

 

cool cool cool

 Apr 6, 2024
 #4
avatar+16 
-1

"Can you solve these?"<- I think you misspelled "solve these problems for me so I can please the answer box." Quite an egregious error if you ask me!

 Apr 7, 2024
edited by Holtran  Apr 7, 2024

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