hello, i'd be so grateful if anyone could help me with these homework
questions. i'm stuck on a few, and i don't really know where to even
begin with them. they're due in tomorrow so i'm pretty desperate.

Solve the equations:

1) 5^X+2 = 3^2X
2) 0.2E^2X -4.7e^x + 0.3 = 0

[i'm pretty sure that you need to use exponential/log rules but i can't do it]

3) Find the coordinates of the points on the curve y = x^3 - 2x^2 - 3x
+ 1 where the tangent is parallel to the straight line 4x - y = 7.

4) For ideal gases, Boyles' law states that pressure is inversely
proportional to volume. A more realistic relationship between pressure
P and volume V is given by van der Waals equation:

P = a/V^2 + C/V-b

where C is the constant of proportionality, a is a constant that
depends on molecular attraction, and b is a constant that depends on
the size of the molecules. Find the compressibility of the gas, which
is measured by dV/dP.

[i'm not even sure which one of the algebraic/calculus rules i have to
use for this!]

5) For taxation purposes, the formula V = 10^6 x 10^-0.0209t describes
the decrease in value V of a city building t years after its
construction. In how many years will the building be valued at 75% of
its original value?


that's all! thankyou for reading, i really would appreciate ANY help
you can give me. i'd love to be able to understand them and get them
all right as a big test is coming up. :)

Question 2: 0.2E^2X -4.7e^x + 0.3 = 0

Assume E^x = A
So the question can be rewritten as 0.2A^2-4.7A+0.3 = 0

Multiple by 10 : 2A^2 -47A +3 =0
From the formula  [-b +(b^2-4ac)^0.5]/2a 

You can find A = 0.064 and 23.436

So x can be calculate using Log rule then x = ln(A) = -2.748 and 3.154

thankyou so much ! :) that makes sense now. it really really helps. i
don't suppose you know how to work out any of the others? lol.

Question 1

x= 0.77366878389

Question 5
V = 10^6 x 10^-0.0209t 

Assume T at start = 0 year Then V(start) = 10^6

So we want T at V = 0.75x(10^6)

Therefore  0.75x(10^6) = (10^6)x(10^-0.0209t)
then ln(0.75) = -0.0209t x ln(10)  ----->  t = 13.76


Additional for question 1, use the calculator to plot the graph.
y=3^2X -5^X -2 and y=0 the intersection is the answer.

wow! i can't thank you enough! you've been extremely helpful..how come
you're not a researcher with google? i feel bad that you can only
comment and not get paid..

Question 4: P = a/V^2 + C/V-b

Differentiate dP/dV = -2a(V^-3)  -c(V^-2) 
Then dV/dP = (-2a-cV)/(V^3)

Question 3: 
The required slope is calculated from y = 4x - 7, so the slope equal 4.
Differentiate the first equation, you will get the slope equation and
make it equal 4. Then you can solve for value of x.

3x^2 -4x -3 = 4  ----> (3x-7)(x+1) = 0

Then the coordinate you want is (2.33,-5.18) and (-1,0)


Good luck for your exam. I hope I answer all of your questions.

sorry, just wondering how you got -5.18 for the y co-ordinate. i
substituted 2.33 into the original equation and i keep getting -4.18
if i plus the 1 on the end.

Haha.. you are correct. I didn't see +1 in that formula. :) it should
be (2.33,-4.18) and (-1,1)