"Phoebe Small has 35 pages of history to be read after she has been reading for ten minutes and 5 pages left after she has been reading for 50 minutes. Assume that the number of pages left to read varies linearly with the number of minutes she has been reading.
A) Write and the linear model that represents how many pages left to read and minutes read.
B) How many pages does she have to read in all?
C) When does she finish?"
The way the problem is stated tells us that y = # of pages left to be read is the dependent variable and x = # of minutes spent reading is the independent variable
This amounts to having ordered pairs of (# min reading, # pages left to be read)
So you have (10,35) and (50, 5). You can now find the slope m
m = (y2-y1)/(x2-x1) or (35-5)/(10-50) = 30/-40 = -3/4
You can write the equation in slope intercept form by first using the point slope formula of
y-y1=m(x-x1), either of the two points you have can be used, I will use (10,35):
y-35 = -3/4(x-10)
y-35 = (-3/4)x + 30/4 now solve for y to put the equation in slope intercept form
y-35+35 = (-3/4)x + 30/4 + 35
y = (-3/4)x + 30/4 + 140/4
y = (-3/4)x + 170/4
We don't know how many pages Phoebe already read after 10 minutes when she has 35 more pages to read so let a = the number of pages Phoebe had read in 10 minutes. Write as a ratio of #pages/#minutes -> a/10
She read for a total of 50 minutes but since she read from the 10 minute mark to the 50 minute mark which is a difference of 40 minutes we know that she read 30 pages (because she went from 35 pages to having only 5 pages left in 40 minutes thus that ratio is 30/40
Set up the proportion a/10 = 30/40 solve for a = 30/4 which is 7.5 pages, thus she read a total of 35 pages + 7.5 pages ~ 43 pages in all
*Or you know the rate at which Phoebe was reading was 3 pages every 4 minutes and that she read for 10 minutes which is 10*3/4 = 304/ = 7.5
Since y is the number of pages to be read, when y = 0 the pages are read.
0 = #-3/4)x+ 170/4 subtract 170/4 from both sides
-170/4 = (-3/4)x divide both sides by -3/4
170/3 = x and 170/3 is 56 2/3 minutes, so she finishes reading in 56 2/3 minutes