exponents with same base

[Duffett.matt]

The temperature inside a certain industrial machine at time t seconds after startup, for 0 < t < 10, is given by h(t) = 42t + 1 – 4t + 2 degrees Celsius. How many seconds after startup is the temperature inside the machine equal to 128 degrees Celsius?

I set 128 equal to the equation... the equation should read: h(t) = 4 raised to the 2t + 1..... minus.... 4 raised to the t + 2

I made 128 equal to this, reduced them all down to bases of 2 and ended up with

7= 4t + 2 - (2t + 4)

[tim]

You can't just strip out the 2's in this case since you're adding two different powers on the right side of the equation. What you want to do is this:

128 = 4^(2t + 1) – 4^(t + 2)

pull out the constants:
128 = 4*(4^t)^2 - 16*4^t

divide everything by 4:
32 = (4^t)^2 - 4*4^t

now substitute x for 4^t:
32 = x^2 - 4x
x^2 - 4x - 32 = 0
(x-8)(x+4) = 0
x = 8 or -4
4^t = 8 or -4

but 4^t can never be negative, so 4^t = 8. at this point you can turn things into powers of 2:
4^t = 8
2^(2t) = 2^3
2t = 3
t = 3/2

the key on this one is to recognize that this is actually a quadratic. you can notice this because of the 4^t and 4^(2t). of course, you can also guess and check on this one; after you try 1 and 2 you should realize that the answer is between those numbers, and 1.5 is the next logical thing to try.

[RonPurewal]

please search the forum before posting. thank you.

the-temperature-inside-a-certain-industrial-machine-t24261.html