Alan

Here's a possible derivation:

You can construct two equations from this as follows:

Let b be the original number of blue marbles and y the original number of yellow marbles.

Then  \(\frac{b}{y}=\frac{4}{3}\\\frac{b+5}{y-3}=\frac{7}{3}\)

Solve these simutaneous equations for b.

For polynomials with repeated roots it can be difficult to identify all the factors.  e.g.  y = x² - 4x + 4   (which is y = (x-2)² ).

Young’s modulus is stress divided by strain.

Stress is force per unit (cross-sectional) area.  

You are given the force.

Choose a diameter from the range given (at least, I assume that’s what you are meant to do as you give no other information). From the diameter, d, calculate the area, A = pi*d²/4.

Hence calculate the stress.

Strain is extension divided by original length.

Choose an extension from the given range (which is presumably in mm).

Choose an original length from the range given. 

From these get the strain.

Divide the stress by the strain to get Young’s modulus.

Note that the result will have units of whatever units you choose for force divided by the square of the units you choose for the diameter (strain is dimensionless).

For constant acceleration problems use one or more of the following equations, as appropriate:

v = u + at

v² = u² + 2as 

s = ut + (1/2)at²

where u = initial velocity, v = final velocity, a = acceleration, t = time taken, s = distance travelled.

No. The correct answer is A.  Look at the axes labels.

Try the following site: https://www.mathsisfun.com/algebra/partial-fractions.html

Incidentally, I’ve just noticed that in my original reply I had (s+1) in the denominator rather than (s-1). However, the same approach should be used, I.e. expand using partial fractions, then inverse Laplace each of the simpler fractions so obtained.

"how do you find the inverse Laplace transform of 5/(s-1)(s2+4)"

Assuming the function of interest is  \(f(s)=\frac{5}{(s+1)(s^2+4)}\)

First expand as partial fractions:  \(f(s)=\frac{1}{s-1}-\frac{s}{s^2+4}-\frac{1}{s^2+4}\)

Then do the inverse Laplace transforms of these (the first term wil result in an exponential, the second will have the form of a cosine, and the third the form of a sine).

For 1:

\(a = -2\sin(\frac{-3\pi}{12})+1\)   Now a can be calculated in a straightforward way.

and   

\(1=-2\sin(3b)+1\)   Now rearrange to find b.

I'm sure you can find a and b from these (note that there will be an infinite number of values of b).

Use the same approach for questions 2 and 3.

"Suppose a single bacterium is placed in a bottle at​ 11:00 am. It grows and at​ 11:01 divides into two bacteria. These two bacteria each grow and at​ 11:02 divide into four​ bacteria, which grow and at​ 11:03 divide into eight​ bacteria, and so on.​ Now, suppose the bacteria continue to double every minute and the bottle is full at​ 12:00. How many bacteria are in the bottle at​ 11:39? What fraction of the bottle is full at that​ time?"

After 1 min there are 2¹ bacteria.

After 2 mins there are 2² bacteria.

After 3 mins there are 2³ bacteria.

...

After n mins there are 2ⁿ bacteria.

So after 39 mins there are 2³⁹ bacteria

After 60 mins there are 2⁶⁰ bacteria

Fraction full at 11.39 is thereferore 2³⁹/2⁶⁰ or  2^-21

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