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An equilateral triangle with side t has the same area with a

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An equilateral triangle with side t has the same area with a [#permalink] New post 24 Aug 2008, 19:38
#1. An equilateral triangle with side t has the same area with a square with side s. What is the ratio t:s?

#2. If (2^x)(3^y) = 288 where x and y are positive integers then (2^x-1)(3^y-2) =
16 24 48 96 144

#3. The area of a square garden is A square feet and the perimeter is P feet. If A=2P+9, what is the perimeter of the garden in feet?

28 36 40 56 64

#4. For any positive integer n, the length of n is defined as the number of prime factors whose product product is n. For example the length of 75 is 3 since 75+3*5*5. How many two digit positive integers have length 6?
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Re: 4 questions for you [#permalink] New post 24 Aug 2008, 20:18
pattydread wrote:
#2. If (2^x)(3^y) = 288 where x and y are positive integers then (2^x-1)(3^y-2) =
16 24 48 96 144


(2^x)(3^y) = 288
[(2^x)(3^y)]/(2x3x3) = 288/2x3x3
2^(x-1) 3^(y-2) = 16
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Re: 4 questions for you [#permalink] New post 24 Aug 2008, 20:25
pattydread wrote:
#1. An equilateral triangle with side t has the same area with a square with side s. What is the ratio t:s?


area of the equilateral triangle = area of the square
sqrt3/4 (t) = s^2
[sqrt3/4][t^2] = s^2
sqrt3 (t^2) = 4s^2
3^(1/4) t = 2s
t/s = 2/3^(1/4)
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Re: 4 questions for you [#permalink] New post 24 Aug 2008, 20:33
pattydread wrote:
#3. The area of a square garden is A square feet and the perimeter is P feet. If A=2P+9, what is the perimeter of the garden in feet?

28 36 40 56 64


A = sxs
P = 4s

A = 2p + 9
2 (4s) + 9 = sxs
8s + 9 = s^2
s^2- 8s - 9 = 0
s^2- 9s + s - 9 = 0
(s-9) (s-1) = 0
s = 1 or 9

s should be 9 as 1 doesnot satisfy.
p = 36
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Re: 4 questions for you [#permalink] New post 24 Aug 2008, 20:54
#1. An equilateral triangle with side t has the same area with a square with side s. What is the ratio t:s?

=> 1/2*t* v[ t^2 - (t/2)^2 ] = s2
=> 1/2*t*t*v[3]/2 = s^2
=> (t/s)^2 = 4/v[3]
=> t/s = 2/(3)^(1/4)


#2. If (2^x)(3^y) = 288 where x and y are positive integers then (2^x-1)(3^y-2) =
16 24 48 96 144

=> (2^x-1)(3^y-2) = (2^x)(3^y) / (2*9) = 288/18 = 16

#3. The area of a square garden is A square feet and the perimeter is P feet. If A=2P+9, what is the perimeter of the garden in feet?

28 36 40 56 64

=> s^2 = 2*4s + 9
=> (s-9)(s+1)= 0 ; as side can not be -ve S=9 , so perimeter = 4*S= 36

#4. For any positive integer n, the length of n is defined as the number of prime factors whose product product is n. For example the length of 75 is 3 since 75+3*5*5. How many two digit positive integers have length 6?

=> 2^6 = 64 and 2^5*3= 96 => so total 2 double digit numbers with length 6
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Re: 4 questions for you [#permalink] New post 24 Aug 2008, 21:59
pattydread wrote:
#4. For any positive integer n, the length of n is defined as the number of prime factors whose product is n. For example the length of 75 is 3 since 75=3*5*5. How many two digit positive integers have length 6?


only 2 two digit integers have length 6

count 6 primes.

2x2x2x2x2x2 = 64
2x2x2x2x2x3 = 96
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Re: 4 questions for you [#permalink] New post 26 Aug 2008, 07:45
For question number 2 i was trying to determine values for x and y and then solve for (2^x-1) (3^y-2). How do u determine that u should start by dividing by 2*3*3?

For instance if we divide by 2*3 we get 48 which is one of the answer choices.
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Re: 4 questions for you [#permalink] New post 26 Aug 2008, 09:59
pattydread wrote:
#1. An equilateral triangle with side t has the same area with a square with side s. What is the ratio t:s?

#2. If (2^x)(3^y) = 288 where x and y are positive integers then (2^x-1)(3^y-2) =
16 24 48 96 144

#3. The area of a square garden is A square feet and the perimeter is P feet. If A=2P+9, what is the perimeter of the garden in feet?

28 36 40 56 64

#4. For any positive integer n, the length of n is defined as the number of prime factors whose product product is n. For example the length of 75 is 3 since 75+3*5*5. How many two digit positive integers have length 6?


1)

Let t = 1
Height of triangle = sqrt(1^2 - (1/2)^2) = sqrt(3/4) = sqrt(3)/2
Area of triangle = (1/2)*1*sqrt(3)/2 = sqrt(3)/4

Area of square = sqrt(3)/4 = s^2
s = (3^(1/4))/2

t:s = 1/(3^(1/4)/2 = 2/(3^(1/4))


2) 288 = 144*2 = 12*12*2 = 3*4*3*4*2 = (3^2)*(2^5)

(2^(x-1))*(3^(y-2)) = (2^4)*(3^0) = 16


3) Let s = side of garden

A = 2P + 9
s^2 = 2(4s) + 9
s^2 - 8s - 9 = 0
(s - 9)(s + 1) = 0
s = 9 or -1

P = 4*9 = 36

4) 2

Smallest positive integer with length 6 = 2*2*2*2*2*2 = 64
Second smallest integer of length 6 = 2*2*2*2*2*3 = 96
Third smallest integer of length 6 = 2*2*2*2*3*3 = 144
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Re: 4 questions for you [#permalink] New post 26 Aug 2008, 10:01
pattydread wrote:
For question number 2 i was trying to determine values for x and y and then solve for (2^x-1) (3^y-2). How do u determine that u should start by dividing by 2*3*3?

For instance if we divide by 2*3 we get 48 which is one of the answer choices.


2^(x-1) = (2^x)*(2^(-1)) = 2^x / 2
3^(y-2) = (3^y)*(3^(-2)) = 3^y / 3^2
Re: 4 questions for you   [#permalink] 26 Aug 2008, 10:01
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