Sir-Emo-Chappington

Well presumably 43.5 is the reading you got, and there are no further 0s in the reading.

As a result, we can assume at least one of the inaccuracies, which is the final digit is only accurate to ±0.5, giving this an inaccuracy of presumably ±0.05_metres. Of course other errors need to be accounted for so this is not the only one present.

If you mean the sine of -1, just enter "Sin(-1)".

If you mean how to do the inverse sine (Commonly denoted by "Sin^-1()"), enter "asin()".

Add "a" before the trigonomic function to give it's inverse: "asin", "acos", "atan"

! is a common notation to mean "Factorial".

The factorial of x is equivalent to [x] * [x-1] * [x-2] * [x-3] ... [x-n]

where [x-n] = 1

So say I want the factorial of 10.

This is 10 * 9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1

And if I wanted 73!:

73! = 73 * 72 * 71 * 70 * 69 * 68 ... * 3 * 2 * 1

U = 0 The initial speed

V = 24 The final speed

a = 9.807 The acceleration

Using these we have 2 main methods of finding the distance it's fallen (and hence the height of the cliff).

First we can use a "Suvat" equation (Named such for using the 5 values: U, V, a, s and t). I won't get into quite why this equation works.

V² = U² + 2as

Re-arranging it gives:

s = (V² - U²) / 2a

Put in the numbers we got...

s = (24² - 0²) / (2 * 9.807)

s = 576 / 19.614

s = 29.3667788314469257

s = 29.37_metres (to 2 d.p.)

Alternatively, find "time" needed to fall and use that.

Since all objects accelerate at 9.8_ms^-2 regardless of mass, we know that this drop took 24/9.807 = 2.4472_seconds (to 4 d.p.).

We can then put it through this other "suvat" equation:

s = [(U + V) / 2] * t

s = [(0 + 24) / 2] * 2.4472

s = 29.37_metres (to 2 d.p.)

To quickly explain:

On average, it's fallen at (24 + 0)/2 = 12_ms^-1

So the distance fell was 12 * 2.4472 = 29.37_metres (to 2 d.p.)

This is due to a readability issue. You see, normally you follow an order of significance: first you do powers, then you do multiplications, then additions (and the inverse of each happens at the same time, and goes from left to right).

In the first case, you got (-4 * x)^0

Well anything to the power of 0 = 1 so clearly this equation is: (-4 * x)^0 = 1

In the second case, you got -4 * x ^0

The way this must be read is doing the powers first, so we get -4 * 1

This then multiplies and gives us: -4 * x ^0 = -4

Remember: it never hurts to use more parenthesies if you really want to make it clear what order you meant. It's important to make sure the equation is easy to read.

It appears the common ratio is -¹/₅

The sum of a geometric series is:

$${a}{\left({\frac{\left({\mathtt{1}}{\mathtt{\,-\,}}{{\mathtt{r}}}^{{\mathtt{n}}}\right)}{\left({\mathtt{1}}{\mathtt{\,-\,}}{\mathtt{r}}\right)}}\right)}$$

Where:

a = first term

r = common ratio

n = Number of terms of sequence

So the sum of the first 7 terms is:

75 * (1 - (-¹/₅)⁷) / (1 - -¹/₅)

= 75 * (1 - -0.0000128) / (1 + ¹/₅)

= 75 * 1.0000128 / 1.2

= 62.5008

You can't possibly have "x = 9x" unless x is 0.

9x / (5*4) can be re-written:

9x / (5*4)

= 9x / 20

= (9/20) x

= 0.45x

2r * (4r - 8)

Firstly expand the equation.

= 2r*4r - 8*2r

Get the products of the constants

= 8r*r - 16r

Now the product of the "r"s

= 8r² - 16r

(^x/₃) + (^3x/₄) = 2

Well first let's make these fractions have the same denominator  

(^4x/₁₂) + (^9x/₁₂) = 2

Now combine the fractions together

^13x/₁₂ = 2

Multiply both sides by 12 to remove the fraction

13x = 12 * 2

13x = 24

Divide both sides by 13 so "x" is the subject

x = ²⁴/₁₃

x = 1.846 (to 3 d.p.)