All Answers 356007 Answers

If the half life is specified then the decay constant will be (mathematically) irrational because of the ln(2). However, there is no compulsion to specify the half-life; one may specify the decay constant instead.

For real-world situations, both the half-life and the decay constant will only be approximate anyway (because based on experimental measurement with attendant inaccuracies).

I guess the examples provided by the questioner here are artificial examples dreamt up by a teacher.  

Alan: Is the constant of decay ALWAYS a rational number in these examples given by this questioner?: 1/29, 1/719, 1/409.........etc.???!!!. I  thought when you obtain the constant of decay by taking the natural log of 2 or 1/2 and multiplying it or dividing it by half-life, you automatically get an irrational constant of decay. No?.

You are very welcome :))

Went through it with my tutor and he told me it was correct, as well as explained me as to why it was. Therefore, I've learned something, thanks to you. Hope you got a great day in front of you

A (-1 -4) B (-5 -1) what is f(x) ?

I have my own method with these.  It is slightly mathematically floored but it replaces a few different formulas

\(\begin{align*} gradient&=gradient\\ &\\ \frac{y-y_1}{x-x_1}&=\frac{y_2-y_1}{x_2-x_1}\\ &\\ \frac{y--4}{x--1}&=\frac{-1--4}{-5--1}\\ &\\\frac{y+4}{x+1}&=\frac{-1+4}{-5+1}\\ &\\\frac{y+4}{x+1}&=\frac{3}{-4}\\ &\\-4(y+4)&=3(x+1)\\ &\\-4y-16&=3x+3\\ &\\0&=3x+4y+16+3\\ &\\3x+4y+19&=0\\ \end{align*} \)

144/27.5 = 5.236 (rounded up, full result: 5.2363636363636364)

7 x 5.236 = 36.652 cm.

I assume that this is supposed to be a joke, if so, then it is quite poor. 

- JP.

You got it - Sorry for my bad explanation.

Could be lots of things.

Do you want the equartion of the LINE that goes through those 2 points?

US Version:

891 Trillion

189 Billion

643 Million

641 Thousand

and

755.

UK Version:

891 Billion

189 Milliard

643 Million

641 Thousand

and

755.

Question 27.

I am looking at an alternative to Heureka's and Alan's answers.  

Heureka And Alan did both inspire this answer.    Thanks  guys 

I cannot remember all theorems relating to circes so I am going to use this more common theorem.

The angle between a tangent and a chord is equal to the angle subtended by the chord in the alternate segment.

Proof is here:

1U = 0.8

I assume this means f(1/g(x)) → 4*(1/3x) - 5 → 4/(3x) - 5

.

Nowhere does the original question call 719 the half-life.  

The structure of the equation implies that 1/719 is the decay constant, so the half-life would be ln(2)/(1/719) or just 719*ln(2).

See:  http://web2.0calc.com/questions/m-t-85e-t-29-describes-the-amount-of-a-radioactive-isotope-as-a-function-of-time-t-years-after-how-long-the-amount-has-dropped-to-1-10-of#r9

Hmm!

Radioactive decay equation often written as:  \(N=N_0e^{-kt}\) where k is the decay constant.

When \(N=\frac{N_0}{2}\) we have \(\frac{1}{2}=e^{-k\tau}\) where \(\tau\) is the half-life.

Taking logs we get \(\ln{\frac{1}{2}}=\ln{e^{-k\tau}}\)  or \(-\ln2=-k\tau\)  so the relation between half-life and decay constant is \(\tau=\frac{\ln2}{k}\)  or \(k=\frac{\ln2}{\tau}\)

solve the equation 4x-5=11x+16

Subtract 4x from both sides:

-5 = 7x + 16

Subtract 16 from both sides:

-21 = 7x

Divide both sides by 7:

-3 = x    (or just x = -3)

.

Take logs of both sides:

ln(1.6491633333)=ln((1.023)^2n)  where ln represents log to the base e

Now use the property  ln(x^a) = a*ln(x) to get;

 ln(1.6491633333)=2n*ln(1.023)

Rearrange:  n = ln(1.6491633333)/(2*ln(1.023))

You can use a calculator to evaluate this.

fully simplify (4a-b)^4

\(\begin{array}{|rcll|} \hline && \mathbf{ (4a-b)^4 }\\ &=& (4a)^4 - 4\cdot(4a)^3\cdot b^1 + 6\cdot (4a)^2\cdot b^2 - 4\cdot (4a)^1\cdot b^3 + b^4 \\ &=& 4^4a^4 - 4\cdot4^3a^3\cdot b + 6\cdot 4^2a^2\cdot b^2 - 4\cdot 4a \cdot b^3 + b^4 \\ &=& 256a^4 - 4^4a^3\cdot b + 6\cdot 16a^2\cdot b^2 - 16a \cdot b^3 + b^4 \\ &\mathbf{=}& \mathbf{256 a^4 -256 a^3 b+96 a^2 b^2-16 a b^3+b^4 } \\ \hline \end{array}\)