Question 1146012: At A water tank is being filled by water being pumped into the tank at a volume given by the formula, P(t) = 112t +2000 gallons per minute, where t is in minutes. At the same time the water tank has a leak and the volume of water draining out of the tank is given by the formula L(t) = 15t2 gallons per minute, where t is in minutes.
a. The volume, V, of water in the tank at any minute, t, is the difference of the volume of the water being pumped into the tank and the volume of water leaking out of the tank. Find the volume function, V(t).
b. The volume function is a quadratic function and so its graph is a parabola. Find the vertex of the volume function V(t). (Round answer to 1 decimal place) Show work.
c. Interpret the vertex in the context of the problem.
d. At what time (t > 0) will the tank be empty? (Round answer to 1 decimal place)
Found 2 solutions by ankor@dixie-net.com, ikleyn:
Answer by ankor@dixie-net.com(22740)   (Show Source):
You can put this solution on YOUR website! At A water tank is being filled by water being pumped into the tank at a volume given by the formula, P(t) = 112t +2000 gallons per minute, where t is in minutes. At the same time the water tank has a leak and the volume of water draining out of the tank is given by the formula L(t) = 15t^2 gallons per minute, where t is in minutes.
a. The volume, V, of water in the tank at any minute, t, is the difference of the volume of the water being pumped into the tank and the volume of water leaking out of the tank.
Find the volume function, V(t).
V(t) = (112t+2000) - 15t^2
V(t) = -15t^2 + 112t + 2000
b. The volume function is a quadratic function and so its graph is a parabola. Find the vertex of the volume function V(t). (Round answer to 1 decimal place) Show work.
The vertex is on the axis of symmetry, x= -b/2a find that
t = 
t = 3.73 min
Find the volume when t= 3.73
v(t) = -15(3.73^2) + 112(3.73) + 2000
V(t) = -15(13.94) + 417.76 + 2000
V(t) = 2209.1 gal
Vertex, 3.73, 2209.1
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c. Interpret the vertex in the context of the problem.
Water level increases until 3.73 sec, then starts decreasing
:
d. At what time (t > 0) will the tank be empty? (Round answer to 1 decimal place)
Volume in the tank goes to 0
-15(t^2) + 112t + 2000 = 0
Use the quadratic formula; a=-15, b=112, c=2000
I got a positive solution of
t = 15.9 min is will be empty
:
:
Graphically
Answer by ikleyn(52847)  (Show Source):
You can put this solution on YOUR website! .
To the author of this problem / post.
There is a FATAL discrepancy / inconsistency in terminology you use.
When you describe function P(t), you give its dimension unit as "gallons per minute".
Such a function is (has the technical name) "rate of inflow".
But if it is "rate of inflow", it is NOT the volume of the water in the tank.
The same thing is with the function L(t).
When you describe the function L(t), you give its dimension unit as "gallons per minute".
Such a function is (has the technical name) "rate of emptying", or "rate of drainage", or "rate of outflow".
But if it is "rate of emptying", it is NOT the volume of the water in the tank.
Using correct terminology is a MUST when you formulate a problem.
Therefore, there is no any sense in solving the problem as it is presented in the post, until it is fixed.
Make all necessary corrections, and then re-post the problem to the forum.
Do not post it to me personally.
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By the way, when you re-think the problem to the end, you may find that it EITHER has no sense at all,
OR has totally different meaning than you thought before.
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