EMAT 6600
Problem Solving
Grass Consumption by Oxen
Problem: If 6 acres
of grass, together with what grows on the 6 acres during the time of grazing,
keep 16 oxen for 12 weeks, and 9 acres keep 26 oxen 9 weeks, how many oxen will
15 acres keep 10 weeks, the grass growing uniformly all the time?
Solution: Consider
the following reasoning: we have 6
acres of grass for 12 weeks and during these 12 weeks the grass grow
uniformly. So, we have what we started
with and what has grown for 12 weeks.
With that in mind, 16 oxen are fed for these 12 weeks. Let us set up an equation for the
statement. Let g be the amount of grass
we initially have on the 6 acres, h be grass grown per acre per week, and k be
amount of grass eaten per ox per week.
So, we have
6g + 6*12h = 12*16k
Similarly, we have
9g + 9*9h = 9*26k
and 15g + 15*10h = x*10k, where x is the unknown amount of oxen fed on 9 acres
for 10 weeks.
Now, we have three
variables equations. Now we solve for
x.
If we set up the
equation as follow
6g + 6*12h = 12*16k
9g + 9*9h = 9*26k
15g + 15*10h = x*10k
Use the first two
equations to solve for one of the variables.
In this case, it is g.
g = 32 k – 12h. by substitution, h = 2k/3. With the
knowledge of two of the variables, we can now substitute in the third equation.
15(32 k – 12h)+
15*10h = x*10k
480k – 180h +
150h = x*10k
480k – 30h = x*10k
480k – 30(2k/3) = x*10k
460k = x*10k
46 = x
Thus, it takes 46
oxen 10 weeks to consume 15 acres of grass.
Similarly, we solve the following problem assuming that the grass is growing uniformly.
If 12 Oxen eat up 3
1/3 acres of pasture in 4 weeks, and 21 oxen eat up 10 acres of like pasture in
9 weeks; to find how many oxen will eat up 24 acres in 18 weeks.
10g/3 + 40h/3 = 48k
10g + 90h = 189k
24g + 432h = 18xk
Hence, we have
g = 144k/10 - 40h/10 and h = 9k/10.
By substitution, we
have 36 oxen to eat up 24 acres in 18 weeks.
*About if the grass
has reached its max growth, that is to say that the grass is no longer growing
as the oxen eat for the period given?
Consider the
equations
6g + 12h = 12*16k
9g + 9h = 9*26k
15g + 10h = x*10k