Polluted water flows into a pond. The concentration of pollutant in the pond at the t minutes is modelled by the equation c(t)= 9-90,000(1/10000+3t), where c is measured in kilograms per cubic metre.
a) When will the concentration of pollutant in the pond reach 6 kg/m^3?
b) What will happen to the concreatration of pollutant over time?
so , assuming my guess was correct .....
6 = 9-90,000(1/ (10000+3t) )
-3 = - 90,000(1/ (10000+3t) )
3(10000 + 3t) = 90000
30000 + 9t = 90000
9t = 60000
t = appr 6666.7 minutes or appr 111.1 days
for the second part look at the term
90,000(1/ (10000+3t) )
as t gets larger, the denominator gets larger and as a result 1/ (10000+3t) gets smaller and smaller and eventually approaches zero.
So
-90,000(1/ (10000+3t) ) will get smaller and smaller and you are left with
c(t)= 9 - (very very small)
so the concentration will settle in at 9 kg/m^3
Is it not 4.6 days?
For: t = appr 6666.7 minutes or appr 111.1 days
I have a feeling that there are some brackets in there
e.g. is it
c(t)= 9-90,000(1/ (10000+3t) )
check and come back
yea there is, i forgot to add it in there.
a) Well, to find out when the concentration of pollutant in the pond reaches 6 kg/m^3, we can simply plug in 6 for c(t) in the equation and solve for t:
6 = 9 - 90,000(1/10,000 + 3t)
Let's do the math and see what we get!
b) As for what will happen to the concentration of pollutant over time, let's hope it decreases because nobody wants a pond full of pollution. But hey, maybe the fish will start throwing parties and call themselves the "Toxic Tadpoles." It could be a whole new underwater ecosystem!
To find the time when the concentration of pollutant in the pond reaches 6 kg/m³, we need to solve the equation c(t) = 6.
a) Solving the equation for t:
c(t) = 6
9 - 90,000(1/10,000 + 3t) = 6
Simplify:
-90,000(1/10,000 + 3t) = 6 - 9
-90,000(1/10,000 + 3t) = -3
Divide by -90,000:
1/10,000 + 3t = 3/90,000
Simplify:
1/10,000 + 3t = 1/30,000
Subtract 1/10,000:
3t = 1/30,000 - 1/10,000
Simplify:
3t = (1 - 3)/30,000
3t = -2/30,000
Simplify:
t = -2/90,000
Therefore, the concentration of the pollutant will reach 6 kg/m³ when t = -2/90,000. However, since time cannot be negative in this context, we can conclude that the pollutant will not reach a concentration of 6 kg/m³ in the given scenario.
b) Based on the equation c(t) = 9 - 90,000(1/10,000 + 3t), we can observe that as time (t) increases, the value of 3t also increases. This will result in the value inside the parentheses (1/10,000 + 3t) becoming larger, leading to a decrease in the overall value of c(t).
In other words, as time progresses, the concentration of the pollutant in the pond will decrease according to the given model.