Question

Applying Rate of change
Ellen is downloading a file. After 2 minutes, there are 38.4 megabytes left to download. After 8 minutes, there is only 9.6 megabytes left to download. What is the size, in megabytes, of the file?

Answer 1

$\bf \begin{array}{ccllll} minutes(x)&Mbs\ left(y)\\ -----&-----\\ 2&38.4\\ 8&9.6 \end{array}\\\\ -----------------------------\\\\ slope = {{ m}}= \cfrac{rise}{run} \implies \cfrac{{{ y_2}}-{{ y_1}}}{{{ x_2}}-{{ x_1}}}\impliedby \textit{rate of change} \\\\\\ m=\cfrac{9.6-38.4}{8-2}\implies m=-\cfrac{24}{5}\iff m= -4.8$

now, notice, the rate of change for downloading is negative, because, "y" is decreasing,  namely the Mbs to be downloaded, are less and less and less as the minutes go by, because the file is almost fully downloaded

so is -4.8

now... at 8minutes, there are 9.6Mbs to download
bear in mind that 9.6 is just 4.8 * 2
that simply means, another minute, another 4.8 Mbs, and another minute and another 4.8 Mbs and the file is done

so, the file downloaded really in 10minutes

now, we know the rate is -4.8 or -24/5,   let us nevermind the sign for now

since we know the file is downloading at 24Mbs in 5minutes, a rate of 24/5

how much is it for 10 minutes?   well 10 is really just 5 * 2

so if it downloads at a rate of 24Mbs per 5mins, in 10 minutes it downloaded 24*2 or 48Mbs