Answer:
10.5 in
Step-by-step explanation:
Given
Required
Find length AC
The question is not detailed enough; so, I'll assume that b represents line AC.
Having said that;
We start by multiplying both sides by b
Divide both sides by tan(55)
Find tan(55)
<em>(Approximated)</em>
<em />
Length AC is 10.5
Answer:
f(r) = (x -1)(x -4+√7)(x -4-√7)
Step-by-step explanation:
The signs of the terms are + - + -. There are 3 changes in sign, so Descartes' rule of signs tells you there are 3 or 1 positive real roots.
The rational roots, if any, will be factors of 9, the constant term. The sum of coefficients is 1 -9 +17 -9 = 0, so you know that r=1 is one solution to f(r) = 0. That means (r -1) is a factor of the function.
Using polynomial long division, synthetic division (2nd attachment), or other means, you can find the remaining quadratic factor to be r^2 -8r +9. The roots of this can be found by various means, including completing the square:
r^2 -8r +9 = (r^2 -8r +16) +9 -16 = (r -4)^2 -7
This is zero when ...
(r -4)^2 = 7
r -4 = ±√7
r = 4±√7
Now, we know the zeros are {1, 4+√7, 4-√7), so we can write the linear factorization as ...
f(r) = (r -1)(r -4 -√7)(r -4 +√7)
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<em>Comment on the graph</em>
I like to find the roots of higher-degree polynomials using a graphing calculator. The red curve is the cubic. Its only rational root is r=1. By dividing the function by the known factor, we have a quadratic. The graphing calculator shows its vertex, so we know immediately what the vertex form of the quadratic factor is. The linear factors are easily found from that, as we show above. (This is the "other means" we used to find the quadratic roots.)
Answer:
The minimum possible initial amount of fish:
Step-by-step explanation:
Let's start by saying that
= is the initial number of fishes
John:
When John arrives:
the remaining fish are expressed by the above expression. Let's call it John
and simplify it!
When Joe arrives:
the remaining fish are expressed by the above expression. Let's call it Joe
and simiplify it
since we've already expressed John in terms of x, we express the above expression in terms of x as well.
When James arrives:
We're gonna do this one quickly, since its the same process all over again
This is the last remaining pile of fish.
We know that no fish was divided, so the remaining number cannot be a decimal number. <u>We also know that this last pile was a multiple of 3 before a third was taken away by James</u>.
Whatever the last remaining pile was (let's say ), a third is taken away by James. the remaining bunch would be
hence we've expressed the last pile in terms of n as well. Since the above 'James' equation and this 'n' equation represent the same thing, we can equate them:
L.H.S must be a Whole Number value and this can be found through trial and error. (Just check at which value of n does 2n/3 give a non-decimal value) (We've also established from before that n is a multiple a of 3, so only use values that are in the table of 3, e.g 3,6,9,12,..
at n = 21, we'll see that 2n/3 is a whole number = 14. (and since this is the value of n to give a whole number answer of 2n/3 we can safely say this is the least possible amount remaining in the pile)
by solving this equation we'll have the value of x, which as we established at the start is the number of initial amount of fish!
This is minimum possible amount of fish before John threw out the first fish