Problem 4
Question
The rate that water is entering a tank is given by \(f(t)=40-2 t\) gallons/minute where \(t\) is measured in minutes past noon. (a) Interpret \({ }_{0} A_{f}(x)\) in words, for \(x \geq 0\). (b) For what values of \(x\) is \({ }_{0} A_{f(x)}\) increasing? (c) At what time is the water level in the tank the same as it was at noon? At this time, what is the value of \({ }_{0} A_{f}(x) ?\)
Step-by-Step Solution
Verified Answer
The quantity \({ }_{0} A_{f}(x)\) describes the total volume of water (in gallons) that has entered the tank from noon until \(x\) minutes past noon. \({ }_{0} A_{f}(x)\) is increasing for values of \(x\) less than 20 minutes. The water level is the same as it was at noon at \(x = 20\) minutes past noon, at which time \({ }_{0} A_{f}(x) = 0\).
1Step 1: Interpret \({ }_{0} A_{f}(x)\)
\({ }_{0} A_{f}(x)\) represents the accumulated amount of water from time 0 to time \(x\). It is calculated as the integral of \(f(t)\) from 0 to \(x\). Therefore, in words, \({ }_{0} A_{f}(x)\) represents the quantity of water that has entered the tank from noon until \(x\) minutes past noon.
2Step 2: Determine when \({ }_{0} A_{f}(x)\) is increasing
\({ }_{0} A_{f}(x)\) is increasing when the rate of water entering the tank, which is represented by \(f(t)\), is positive. We set \(f(t) > 0\) which gives us \(40-2t > 0\) or \(t < 20\). Therefore, \({ }_{0} A_{f}(x)\) is increasing for \(x\) values less than 20 minutes.
3Step 3: Find the time when water level is the same as at noon
The water level is the same as it was at noon when the accumulated amount of water is zero. That is, when \({ }_{0} A_{f}(x) = 0\), which happens when the integral of \(f(t)\) from 0 to \(x\) equals 0. Solving the equation, we find \(x = 20\) minutes. At this time, the value of \({ }_{0} A_{f}(x)\) is also zero because the amount of water that has entered equals the amount of water that has left.
Key Concepts
Rate of ChangeIntegral CalculusMathematical InterpretationWater Flow Rate
Rate of Change
The rate of change is a foundational concept in calculus and describes how a quantity varies over time or space. In the context of this problem, the rate at which water enters a tank is given by the function \( f(t) = 40 - 2t \), where \( t \) represents time in minutes past noon. This function is a linear expression, indicating that the water flow rate decreases by 2 gallons per minute as time progresses.
By understanding the rate of change, we can establish that initially, at \( t = 0 \), the rate at which water enters the tank is 40 gallons per minute. Gradually, as \( t \) increases, this rate diminishes, reaching 0 gallons per minute when \( t = 20 \). Thus, the flow rate offers insights into how quickly the tank is being filled or drained at any given moment, depending on the value of \( t \).
By understanding the rate of change, we can establish that initially, at \( t = 0 \), the rate at which water enters the tank is 40 gallons per minute. Gradually, as \( t \) increases, this rate diminishes, reaching 0 gallons per minute when \( t = 20 \). Thus, the flow rate offers insights into how quickly the tank is being filled or drained at any given moment, depending on the value of \( t \).
Integral Calculus
Integral calculus is used to compute the total change or accumulation of a quantity when given its rate of change. In this exercise, to find the total gallons of water that have entered the tank by time \( x \), we utilize the concept of integration.
The notation \( {}_{0} A_{f}(x) \) represents the integral of the function \( f(t) \) from the time \( t = 0 \) to \( t = x \). Formally, it is expressed as:
The notation \( {}_{0} A_{f}(x) \) represents the integral of the function \( f(t) \) from the time \( t = 0 \) to \( t = x \). Formally, it is expressed as:
- \( A(x) = \int_{0}^{x} (40 - 2t) \, dt \)
Mathematical Interpretation
When solving real-world problems using calculus, interpreting mathematical symbols and results is key. In the given exercise, we have the integral expression \( {}_{0} A_{f}(x) \), serving as a measure of the total amount of water entered into the tank up to a specific time \( x \).
Understanding this involves recognizing that when \( {}_{0} A_{f}(x) \) is zero, it implies that the net change in water level is zero, meaning the amount of water that has entered equals that which has left or has not yet been added. To determine when \( {}_{0} A_{f}(x) \) is increasing, we look at when \( f(t) \) is positive, i.e., when water is still entering the tank. This happens as long as \( t < 20 \).
Further analysis shows at what time the water level returns to its original state (noon) which occurs at \( t = 20 \). Thus, mathematical interpretation bridges abstract concepts with practical outcomes.
Understanding this involves recognizing that when \( {}_{0} A_{f}(x) \) is zero, it implies that the net change in water level is zero, meaning the amount of water that has entered equals that which has left or has not yet been added. To determine when \( {}_{0} A_{f}(x) \) is increasing, we look at when \( f(t) \) is positive, i.e., when water is still entering the tank. This happens as long as \( t < 20 \).
Further analysis shows at what time the water level returns to its original state (noon) which occurs at \( t = 20 \). Thus, mathematical interpretation bridges abstract concepts with practical outcomes.
Water Flow Rate
The water flow rate in this problem is dictated by the function \( f(t) = 40 - 2t \). Initially, the rate is quite high, allowing the tank to fill quickly, but it slows down over time. Such an equation helps us predict and understand how quickly the tank is being filled in the first few minutes and how this filling rate changes as time progresses.
Analyzing the water flow rate tells us whether the tank is filling up or eventually stopping. When water flow rates are positive, the tank is actively filling. Conversely, when the rate eventually reaches zero at \( t = 20 \) minutes, the inflow of water ceases. Understanding such expressions aids in planning and managing resources like water more efficiently
- especially in contexts where the management of water supply and storage is crucial, such as agriculture, city planning, or even home use.
Analyzing the water flow rate tells us whether the tank is filling up or eventually stopping. When water flow rates are positive, the tank is actively filling. Conversely, when the rate eventually reaches zero at \( t = 20 \) minutes, the inflow of water ceases. Understanding such expressions aids in planning and managing resources like water more efficiently
- especially in contexts where the management of water supply and storage is crucial, such as agriculture, city planning, or even home use.
Other exercises in this chapter
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