Problem Solving

Could someone please explain how to solve these problems. Thanks

Question 1:

If 3^6x = 8,100, what is the value of (3^(x – 1))^3 ?


A. 90
B. 30
C. 10
D. 10/3
E. 10/9

D

Question 2

On an aerial photograph, the surface of a pond appears as circular region of radius 167 inch. If a distance of 1 inch on the photograph corresponds to an actual distance of 2 miles, which of the following is the closest estimate of the actual surface area of the pond, in square miles?

A. 1.3
B. 2.4
C. 3.0
D. 3.8
E. 5.0

B

Question 3

Six cards numbered from 1 to 6 are placed in an empty bowl. First one card is drawn and then put back into the bowl; then a second card is drawn. If the cards are drawn at random and if the sum of the numbers on the cards is 8, what is the probability that one of the two cards drawn is numbered 5 ?

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

D

Q1: Backsolve.

Start with C. (3^(x-1)) = 3^3x/3^3= 10

which is 3^3x = 90x3

Square both sides

(3^3x)^2 = 8100 x 9
which is 3^6x = 8100x9

Which is more than 8100. Hence the answer should be lower than 10.

Pick D - 10/3. If you solve you will get 3^6x = 8100. Hence answer is D

Q3.

All possible events - 5: (2,6 3, 5 4, 4 5, 3 6, 2)

Desired events - 2: (3, 5 5, 3)

Probability = desired events/possible events = 2/5

answer is D: 2/5

Q2. I am not able to understand the story... The actual radius per scale provided will be 167x2=334 miles. The Surface area of the pond will be a huge number

pixRaidus^2 = 3.14x3.34x3.34x10^4

which is approx 310000 square miles


Not sure where the problem is...

There's a technique to number 1:

3^6x=8100

(3^(x-1))^(3)=(3^x/3)^(3)=3^3x/27

3^3x=The square root of 3^6x= square root of 8100=90

90/27=10/3

truplayer256 wrote:There's a technique to number 1:

3^6x=8100

(3^(x-1))^(3)=(3^x/3)^(3)=3^3x/27

3^3x=The square root of 3^6x= square root of 8100=90

90/27=10/3

Is there a simpler way to tackle this question??
anyone?

fruti_yum wrote:

truplayer256 wrote:There's a technique to number 1:

3^6x=8100

(3^(x-1))^(3)=(3^x/3)^(3)=3^3x/27

3^3x=The square root of 3^6x= square root of 8100=90

90/27=10/3

Is there a simpler way to tackle this question??
anyone?

I felt my solution to be simpler but not sure. Lets see.

3^6x = 8100, so 3^3x = 90.
Because, 8100 = 81 * 100 and to get 3x from 6x, you will have to divide it by 2. Which means a square root.

Now, lets simplify the question asked:
(3^(x-1))^(3) = 3^(3(x-1)) {You might want to write this down on a paper}
Which is nothing but, 3^3x / 3^3 , i.e: 3^3x / 27. {This is the simplified question asked}.

So, 90/27 = 10/3.

Question 1:
Simplify (3^(x - 1))^3 = (3^x*1/3)^3=3^3x* (1/27)
Take the square root from 3^6x = 8,100, thus 3^3x = 90
You will have 90/27=10/3
Answer is D

Question 2
Something wrong with the numbers
check it

Question 3
There are following possible combinations to get 8: 4+4, 5+3, 6+2 and 5 possible variants in total (in 5+3 and 6+2 we can draw 5 and 6 first or second).
5 + 3 gives two possible variants - (5, 3) and (3, 5)
The probability is 2/5
Answer is D