Maths Question Please Help?

Question

Hi, i was wondering if anyone could help me on the following questions at:

http://www.gcsemathspastpapers.com/images/p5j01q49.gif

I really need help with this section for revision (exam on monday) and was wondering if someone could tell me the answer and how they worked it out.


Thanks in advance

Answer 1

OK, let's see what we can do:

sqrt(243) = 3^q. This can be solved using logarithms, but I don't think that's how you want to do it. For the moment, let's square both sides. Thus,

243 = 3^(2q).

Now, 243 happens to be 3^5. So now you can write:

3^5 = 3^(2q)

And thus, 5=2q or q=2.5

Now, the next one:
(sqrt(12)-sqrt(3))^2 = 3^t

Since 12 = 4*3, we can rewrite the left side as:

(2*sqrt(3) -sqrt(3))^2 = 3^t

simplifying the left side, we have

(sqrt(3))^2 = 3^t

and since the left-hand side is equal to 3, we can write:

3 = 3^t

so t=1 (amazing, huh?)

And now the last one:

since the area of a rectangle is length x width, we have:

sqrt(243) = (sqrt(12) - sqrt(3))*3^m

now we know from the first problem that sqrt(2430= 3^2.5, and from the second problem, since

(sqrt(12)-sqrt(3))^2 = 3^1

then

(sqrt(12) - sqrt(3) = 3^0.5 (we just took the square root of both sides)

so the last equation can now be written as:

3^2.5 = 3^0.5 * 3^m

or

3^2.5 = 3^(0.5+m)

so

2.5 = 0.5 + m

and thus

m= 2

Answer 2

√243 = 3^q

0.5 ln 243 = q ln 3

(0.5 ln 243)/(ln 3) = q

q = 2.5


(√12 - √3)² = 3^t

(2√3 - √3)² = 3^t

3 = 3^t

t = 1

Area = length * width

√243 = (√12 - √3)(3^m)

√243/√3 = (3^m)

√81 = 3^m

9 = 3^m

m = 2
.

Answer 3

1) This question depends on you understand of logarithms. If you have forgotten the rules of logarithms, a site like SOS Math might help (http://www.sosmath.com/algebra/logs/log4/log44/log44.html )

sqrt(243) = 3^q
9*sqrt(3) = 3^q
ln(9*sqrt(3)) = ln(3^q)
ln(9) + ln(sqrt(3)) = q*ln(3)
ln(3^2) + ln(3^(1/2)) = q*ln(3) -> Note 1
2*ln(3) + 1/2*ln(3) = q*ln(3)
(2 + 1/2)*ln(3) = q*ln(3)
((2+1/2)*ln(3))/ln(3) = q
2..5 = q

Note 1: we did this because we noticed a ln(3) on the right hand side and will want to get rid of it

2: This question is very similar to the first.

(sqrt(12) - sqrt(3))^2 = 3^t
(2*sqrt(3) - sqrt(3))^2 = 3^t
((2-1)*sqrt(3))^2 = 3^t
sqrt(3)^2 = 3^t
3 = 3^t
ln(3) = t * ln(3)
ln(3)/ln(3) = t
1 = t

3) This last problem is again like the first two.

l*w = sqrt(243)
(3^m)*(sqrt(12)-sqrt(3)) = 9*sqrt(3)
(3^m)*((2-1)*sqrt(3)) = 9*sqrt(3)
(3^m)*sqrt(3) = 9*sqrt(3)
((3^m)*sqrt(3))/sqrt(3) = 9
3^m = 9
m*ln(3) = ln(3^2)
m*ln(3) = 2*ln(3)
m = 2

This type of mathematical notation is hard to read. To fully understand the problems you asked about I recommend that you re-write my solutions by hand in proper notation.

Hope this helps.