Let t be time in seconds and let r(t) be the rate, in gallons per second, that water enters a reservoir:
r(t)=700−40t.
a) For 0≤t≤30, when does the reservoir have the most water?
b) For 0≤t≤30, when does the reservoir have the least water?
if a(t) is the amount of water at time t, and the tank is initially empty,
da/dt = r(t) = 700-40t
a(t) = 700t-20t^2
a(t) is a parabola with vertex at t=700/40=17.5
So, that's the time of max water.
For minimum, find whether a(30) is positive or negative. Recall that a(0) = 0.
To find the time when the reservoir has the most and least water, we need to find the maximum and minimum points of the function r(t) = 700 - 40t.
a) To find the time when the reservoir has the most water, we need to find the maximum point of the function.
Step 1: Take the derivative of r(t) with respect to t.
r'(t) = -40
Step 2: Set the derivative equal to zero and solve for t.
-40 = 0
Since -40 is a constant and does not depend on t, there is no solution for t.
Therefore, there is no maximum point for the function r(t) = 700 - 40t within the given interval 0≤t≤30.
b) To find the time when the reservoir has the least water, we need to find the minimum point of the function.
Step 1: Take the derivative of r(t) with respect to t.
r'(t) = -40
Step 2: Set the derivative equal to zero and solve for t.
-40 = 0
Since -40 is a constant and does not depend on t, there is no solution for t.
Therefore, there is no minimum point for the function r(t) = 700 - 40t within the given interval 0≤t≤30.
To determine when the reservoir has the most and least amount of water within the given time frame, we need to find the points where the rate of water entering the reservoir is zero.
a) When does the reservoir have the most water?
To find the time when the reservoir has the most water, we need to find when the rate of water entering the reservoir is zero, i.e., when r(t) = 0.
Set the rate function equal to zero and solve for t:
700 - 40t = 0
Adding 40t to both sides:
40t = 700
Dividing both sides by 40:
t = 17.5
Therefore, when t = 17.5 seconds, the reservoir has the most amount of water within the given time frame.
b) When does the reservoir have the least water?
To find the time when the reservoir has the least water, we need to find when the rate of water entering the reservoir is maximum.
The rate function is given by r(t) = 700 - 40t. The rate is decreasing with time because the coefficient of t is negative (-40). The maximum rate occurs at the beginning of the time interval (t = 0).
Therefore, when t = 0 seconds, the reservoir has the least amount of water within the given time frame.