+0  
 
+1
88
12
avatar+206 

7.  The numbers 19, a, b, c, 73 form an arithmetic sequence, in that order.
Find a+b+c.
 

8. The number 8!+9!+10! is equal to n*8! for some integer n. What
is the value of n?

 

9. Regular octagon ABCDEFGH has side length 4 units. What is the
area of square ACEG? Express your answer in simplest radical form.

 

10. The least common multiple of 210 and n is 210n, where n is an integer
greater than 1. What is the smallest possible value of n?

 

11. The side lengths of an acute triangle, in inches, are integers and form
an increasing arithmetic sequence. What is the smallest possible value
for the perimeter of the triangle?

 

 

Please answer all questions...Thanks!

mathtoo  Feb 18, 2018
Sort: 

12+0 Answers

 #1
avatar
+1

7)

 

73 -19 =54

54 / 4 = 13.5 - common difference

19, 32.5, 46, 59.5, 73.........etc.

a + b + c = 32.5 + 46 + 59.5 = 138

 

8)

[8! + 9! + 10!] =8! x [(1 + 9 + 90)] / 8! x n

n = 100

Guest Feb 18, 2018
edited by Guest  Feb 18, 2018
edited by Guest  Feb 18, 2018
edited by Guest  Feb 18, 2018
edited by Guest  Feb 18, 2018
 #5
avatar+2592 
+1

Yes, I think both of them are correct...... 

tertre  Feb 18, 2018
 #2
avatar+116 
+1

How did you get 54/3=13.5?

azsun  Feb 18, 2018
edited by azsun  Feb 18, 2018
edited by azsun  Feb 18, 2018
 #3
avatar
+1

Typo !!!!!!!!!?.

Guest Feb 18, 2018
 #4
avatar+116 
+1

Wait till CPhill comes. He's great at this.

azsun  Feb 18, 2018
 #6
avatar+12266 
+1

9.  Interior angles of a regular polyhedron total   (n-2)(180)

      for an octagon    (8-2)(180) = 1080    there are 8 interior angles so each is 1080/8 = 135 degrees

 

     The sides of the SQUARE form isocoles triangles of sides 4, 4   and 'x' (the side of the square) with an included 135 degree angle beteen the 4 and 4 sides

   The other two angles are   (180-135)/2 = 22.5 degrees

 Use the law of sines to find 'x'

sin 135 /x  = sin22.5  / 4    results in square side length of 7.39     

 Square Area = 7.39 x 7.39 =54.63 units^2 

ElectricPavlov  Feb 18, 2018
 #7
avatar+12266 
+2

11.     The smallest I can find ...by trial and error    is    4  5   6

           perimeter = 15

ElectricPavlov  Feb 18, 2018
 #8
avatar+2592 
+2

Yep, that's what I got, too!

tertre  Feb 18, 2018
 #9
avatar+86528 
+1

7.  The numbers 19, a, b, c, 73 form an arithmetic sequence, in that order.
Find a+b+c.

 

19 + 4d  =  73       subtract  19  from both sides

 

4d  =  54   divide both sides by 4

 

d  =  13.5

 

Calling m the first term  ....the sum  of a + b + c  is

 

( m + d ) + (m + 2d) + (m + 3d)  =

 

3m  + 6d  =

 

3(19) + 6(13.5)  =

 

57    + 81  =

 

138

 

 

cool cool cool

CPhill  Feb 19, 2018
 #10
avatar+86528 
+1

10. The least common multiple of 210 and n is 210n, where n is an integer
greater than 1. What is the smallest possible value of n?

 

For  210n to be the LCM, n will be the first positive integer that does not share any factors with 210.....this is 11

 

LCM   210*11  =  2310

 

 

cool cool cool

CPhill  Feb 19, 2018
edited by CPhill  Feb 19, 2018
 #11
avatar+2592 
0

Wow! Great solution!

tertre  Feb 19, 2018
 #12
avatar+86528 
+1

11. The side lengths of an acute triangle, in inches, are integers and form
an increasing arithmetic sequence. What is the smallest possible value
for the perimeter of the triangle?

 

Let m  be the shortest side, m + d the next longest,and m + 2d  the longest

 

And  we have, by the Triangle Inequality :

 

 ( m )  + (m + d) > m + 2d

2m  + d > m + 2d

m > d 

Since we have integers, the smallest possible value of d is 1.....which means the smallest  possible value of m must be 2 

 

This means that  the  the second term, m+ d  ≥ 3  and the third term  m + 2d ≥ 4

 

If the triangle is  2,3,4   we must have the following to be acute

 

4 < √ [ 2^2 + 3^ 2]

4 < √13       which is untrue

 

A  3,4,5 triangle is right

 

If the triangle is  4,5,6  we must have that

6 < √ [ 4^2 + 5^2]

6 < √41  which is true

 

So....4,5,6   is the triangle we want....and the smallest perimeter is  15

 

 

cool cool cool

CPhill  Feb 19, 2018

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