Problem Solving

Hi guys,

Can you help with a few questions here, please?

1&2) As per the attachments.

3) If (cubed)√x-b = c^6, then x = ?

A. c^2 + b
B. c^6 + b
C. c^9 + b
D. c^12 + b
E. c^18 + b

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?

A. none
B. one
C. two
D. three
E. four

What's the quickest way to solve this one and the one below??

5) Connie paid a sales tax of 8% on her purchase. If the sales tax had been only 5%, she would have paid $12 less in sales tax on her purchase. What was the total amount that Connie paid for her purchase including sales tax?

I tried to use this formula: sales tax% = (sales tax/ purchase price)*100 but could not get very far...

6) Machine A produces pencils at a constant rate of 9,000 pencils per hour, and machine B produces pencils at a constant rate of 7,000 pencils per hour. If the two machines together must produce 100,000 pencils and if each machine can operate for at most 8 hours, what is the least amount of time, in hours, that machine B must operate?

A. 4
B. 4 2/3
C. 5 1/3
D. 6
E. 6 1/4

All OAs will be posted. Thanks!

1) IMO D

area of equilateral triangle=sqrt(3)/4 *t^2
area of square=s^2
s^2=sqrt(3)/4 *t^2
t/s=2/sqrt(sqrt(3))

2)IMO E
eq of line is y=-1/2*x+3

so any y below the line will be
<-1/2*x+3.. check with a few points, it satisfies

3)(cubed)&#8730;x-b = c^6
cubing both sides
x-b=c^18
x=c^18+b
IMO E

4)IMO B
only 2^6 is possible

5)IMO 432
if she had paid 5 percent tax..she would have paid 12 dollars less
so 3 percent of the price is 12 dollars

.03X=12
x=400
8 percent of X=32
total=432 dollars

6)IMO A
we want machine B to operate least
if machine A runs for 8 hours, it makes 72000 pencils.
remaining pencils(28000) are to be made by B)
so B works for 4 hours

tohellandback, all perfectly makes sense and the answers match apart from 4)B only 2^6 is possible. The OA is C -> 2 integers. Thanks

LevelOne wrote:tohellandback, all perfectly makes sense and the answers match apart from 4)B only 2^6 is possible. The OA is C -> 2 integers. Thanks

oops!! my mistake..yeah there is another one possible ..2^5*3=96

so answer will be 2.