Firstly, let's create a function of f(t) where t represents the time that has past, and f(t) represents the amount of rainwater. We know that when t=1, then f(t)=10, and t=2 then f(t)=15. So, let's take that and analyze it:
(1,10)
(2,15)
m = (15-10)/(2-1) = 5
y-intercept = 5
∴ f(t) = 5t+5
Now we just evaluate t for 10:
f(10) = (5*10)+5
f(10) = 55
Bacteria growth occurs exponentially; bacteria divides into 2 every t minutes (similar to the penny doubling every day story). If one gets filled, and the contents divides once, there will be enough for 2 bottles; when these two are ready to divide, there will be enough for 4. This growth process begins very slowly - 1, 2, 4, 8, 16, 32; but it soon speeds up greatly, 64, 128, 256, 512, 1024, 2048.
Set the whole expression = to 0 and solve for x.
3x^(5/3) - 4x^(7/3) = 0. Factor out x^(5/3): x^(5/3) [3 - 4x^(2/3)] = 0
Then either x^(5/3) = 0, or 3 - 4x^(2/3) = 0.
In the latter case, 4x^(2/3) = 3.
To solve this: mult. both sides by x^(-2/3). Then we have
4x^(2/3)x^(-2/3) = 3x^(-2/3), or 4 = 3x^(-2/3). It'd be easier to work with this if we rewrote it as
4 3
--- = --------------------
1 x^(+2/3)
Then
4
--- = x^(-2/3). Then, x^(2/3) = (3/4), and x = (3/4)^(3/2). According to my 3 calculator, that comes out to x = 0.65 (approx.)
Check this result! subst. 0.65 for x in the given equation. Is the equation then true?
My method here was a bit roundabout, and longer than it should have been. Can you think of a more elegant (and shorter) solution?
Answer:
Move all terms to the left side and set equal to zero. Then set each factor equal to zero.
x=5,−2
Step-by-step explanation:
I can do math
The length of that arc is (90/360)·2·π·6 = (1/4)·12·π = 3·π ≈ 9.42 units;
