Question

How to solve these questions steps by steps ?

The diagram below shows a straight line graph obtained by plotting `log_10 y` against x. The variables x and y are related by the equation `y = (2w )/ 3 ^ x`, where w is a constant.

a) Express the equation `y = (2w) / 3 ^ x` in the linear form, used to obtain the straight line graph as shown in the figure above.

b) Find the value of w.

Answer 1

`log_10 y = - (log_10 3)x + log_10 (2w)`

`:. w = (log_10 3^5)/(1-1/3^10) ~~ 2.39`

Explanation:

Note that the question indicates that the diagram shows a straight line graph obtained by plotting `log_10 y` against `x`. The diagram label on the `y`-axis is therefore erroneously labelled `x^2y`

We have:

`y = (2w) / 3^x`

Part (A):

Taking base 10 logarithms of the given equation, we get:

`log_10 y = log_10 {(2w) / 3^x}`
`\ \ \ \ \ \ \ \ \ \ = log_10 (2w) -log _10(3^x)`
`\ \ \ \ \ \ \ \ \ \ = log_10 (2w) -x log_10 3`

`:. log_10 y = - (log_10 3)x + log_10 (2w)`

Which is in the form,

`Y = mX + c`

which, is that of a straight line.

Part (B):

Given (from the graph) that the straight line passes through the coordinate `(0,a)` we get:

`:. log_10 a = 0 + log_10 (2w)`
`:. a = 2w`

Given (from the graph) that the straight line passes through the coordinate `(10,b)` we get:

`:. log_10 b = - (log_10 3)10 + log_10 (2w)`
`:. log_10 b = log_10 (2w) - (log_10 3^10)`
`:. b = (2w)/3^10`

We can also evaluate the gradient of the line:

`m = (Delta Y)/(Delta X) = (b-a) / (10-0)`

So therefore:

`-log_10 3 = ((2w)/3^10-2w)/10`

`:. -log_10 3 = 2w((1/3^10-1)/10)`

`:. w((1-1/3^10)/5) = log_10 3`

`:. w(1-1/3^10) = 5log_10 3`

`:. w = (log_10 3^5)/(1-1/3^10) ~~ 2.39`