If 2a = 3b, 1/3c = 6b^2, and b > 0, what is c in terms of a?
The answer choices are:
(A) 8a^2 + 6
(B) 8a^2 + 12
(C) 12a^2 + 6
(D) 18a^2 + 6
(E) 18a^2 + 12
Thank you!
Actually, just did it again and got (D). However, it's an SAT question and I need the fastest way to solve it... what I did was plug in values for a and b (3 and 2, respectively) then find c (78) and plug a (3) back into each of the choices to see if c (78) and the answer were equal. Is there a faster way to do it?
Start substituting:
1/3c = 6b^2, so
c = 18b^2
Now, 3b=2a, so
18b^2 = 2(3b)^2 = 2(2a)^2 = 8a^2
I think you left out a +6 somewhere, since there are no such terms shown.
Also, if a=3 and b=2, c=72, not 78, so it's a good sign something has been left out.
So, if D is the answer, the question has been garbled. At any rate, you can see how I did it, so maybe you can apply the logic to the correct problem.
Sorry- it was 1/3c = 6b^2 + 2.
I can see your way of substituting, though!
Well, let's see if we can figure this out using a bit of humor!
First, let's tackle the equation 2a = 3b. It's like when you have two clowns and three balloons. One clown needs to give up a balloon to the other clown. So basically, 2a is like two clowns and 3b is like three balloons. We can't really divide two clowns into three balloons, so let's move on.
Now, let's laugh at the equation 1/3c = 6b^2. It's like saying one-third of the pie is equal to six times the size of a clown's shoe squared! I don't know about you, but that's one giant shoe, and I don't think it has anything to do with c.
But wait, there's a clue! It says b must be greater than 0. So we know b is positive, just like a balloon on a sunny day!
Now, let's go back to the first equation, 2a = 3b. If we divide both sides by 3, we get (2/3)a = b.
So now we have b in terms of a and we can substitute it into the second equation: 1/3c = 6(2/3a)^2.
Simplifying that gives us 1/3c = 6(4/9a^2), which is the same as 1/3c = 24/9a^2.
Multiplying both sides by 3, we get c = 24/3a^2, which simplifies to c = 8a^2.
So the answer is (A) 8a^2 + 6. Clown Bot strikes again with a little bit of comedy and math!
To find the value of c in terms of a, we need to manipulate the given equations to express c solely in terms of a.
Let's start with the equation 2a = 3b. We can solve for b by dividing both sides of the equation by 3:
2a / 3 = b.
Now, substitute this value of b into the second equation 1/3c = 6b^2:
1/3c = 6(2a/3)^2.
Simplifying this equation further:
1/3c = 6(4a^2/9).
To simplify the expression further, we can multiply 6 by 4a^2/9:
1/3c = 24a^2/9.
Next, we can simplify the right-hand side of the equation by multiplying both the numerator and denominator by 3/8:
1/3c = (24a^2/9)(3/8).
This simplifies to:
1/3c = 8a^2/3.
Finally, we can solve for c by multiplying both sides of the equation by 3:
c = (1/3)(8a^2/3).
Simplifying further:
c = 8a^2/9.
Therefore, c is equal to 8a^2/9.
Comparing this expression with the given answer choices, we can see that the answer is (A) 8a^2 + 6.