Question

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The function f(t) represents water going into a swimming pool with respect to the number of hours(t) water is flowing in where(t) represents time. f(t)=t squared +8t+9 There is a leak in the pool and it’s losing water at a rate represented by d(t). d(t)=t squared +11t+4
a.Write a function w(t) to represent the amount of water in the pool using the two functions.
b. Use the new function to determine if the pool will leak all of the water
c. If the pool will drain of all water, how much time will it take?
d.Will f(t) and d(t) intersect on a graph? Explain what it means if they do.
e. What is the domain of f(t), d(t), and w(t)? Explain your answer.

Answer 1

Explanation:

so, if I understand you correctly and nothing was left out :

f(t) = t² + 8t + 9

d(t) = t² + 11t + 4

just by looking at the 2 functions, it is clear that d(t) creates larger functional values for higher values of t.

because t² = t², but 11t > 8t. and the constants in the end are irrelevant for larger numbers.

so the pool will drain.

a.

w(t) = f(t) - d(t) = t² + 8t + 9 - t² - 11t - 4 = -3t + 5

b.

yes, it will completely drain, as -3t will drive it to 0, and if it was physically possible into the negative ask the way to -infinity.

c.

-3t + 5 = 0

5 = 3t

t = 5/3

it will be completely drained after

1 2/3 hours = 1 hour 40 minutes.

d.

to intersect they need to create the same result for a value of t :

t² + 8t + 9 = t² + 11t + 4

8t + 9 = 11t + 4

-3t + 5 = 0

t = 5/3 or 1 hour 40 minutes.

that means that the filling and the draining are only equal, when there is no water in the pool.

that also means that f(t) is actually calculating how much water there is in the pool, if there would not be no drain, with a starting value of 9.

e.

the domain is the interval or set of all valid input values (here for t).

negative t (hours) don't make any sense for any of the 3 functions.

but f(t) can go on theoretically forever. but usually it's limited by the size of the pool (once it reaches the top, there is no more filling). we just don't know anything about that.

so, the domain for f(t) is

0 <= t < +infinity

d(t) could go on and on as well, as long as there is water in the pool.

so, the domain for d(t) is also

0 <= t < +infinity

but w(t) based on its meaning and definition makes only sense between the start (t = 0) and until the water is fully drained by the combination of f(t) and d(t).

so, the domain of w(t) is

0 <= t <= 5/3

we could define w(t) as segmented function

w(t) = -3t + 5 0 <= t <= 5/3

= 0 t > 5/3

THEN the domain would be again

0 <= t < +infinity