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