Question
(a)-Topic-6.2- Approximating Areas with Riemann Sums
(b)-Topic-5.1- Using the Mean Value Theorem
(c)-Topic-8.2-Connecting Position, Velocity, and Acceleration of Functions Using Integrals
(d)-Topic-8.3- Using Accumulation Functions and Definite Integrals in Applied Contexts
1. A customer at a gas station is pumping gasoline into a gas tank. The rate of flow of gasoline is modeled by a differentiable function f , where f(t) is measured in gallons per second and t is measured in seconds since
pumping began. Selected values of f(t) are given in the table.
(a) Using correct units, interpret the meaning of \(\int_{60}^{135}f(t) dt\) in the context of the problem. Use a right Riemann sum with the three subintervals [60, 90], [90, 120], and [120, 135] to approximate the value of \(\int_{60}^{135}f(t) dt\).
(b) Must there exist a value of c, for 60 < c < 120, such that f ‘(c) = 0 ? Justify your answer.
(c) The rate of flow of gasoline, in gallons per second, can also be modeled by \(g(t)=\left ( \frac{t}{500} \right )cos\left ( (\frac{t}{120})^{2} \right )\) for \(0\leq t\leq 150\) . Using this model, find the average rate of flow of gasoline over the time interval \(0\leq t\leq 150\). Show the setup for your calculations.
(d) Using the model g defined in part (c), find the value of g’ (140). Interpret the meaning of your answer in the context of the problem.
▶️Answer/Explanation
1(a) Using correct units, interpret the meaning of \(\int_{60}^{135}f(t)dt\) in the context of the problem. Use a right
Riemann sum with the three subintervals [60, 90 ,] [90, 120 ,] and [120, 135] to approximate the value of \(\int_{60}^{135}f(t)dt\).
\(\int_{60}^{135}f(t)dt\) represents the total number of gallons of gasoline pumped into the gas tank from time t = 60 seconds to time t = 135 seconds.
\(\int_{60}^{135}f(t)dt\)
\(\approx f(90)(90-60)+f(120)(120-90)+f(135)(135-120)\)
\(=(0.15)(30)+(0.1)(30)+(0.05)(15)=8.25\)
1(b) Must there exist a value of c, for 60 < c< 120 , such that f ′(c) = 0 ? Justify your answer.f is differentiable. ⇒ f is continuous on [60, 120] .
\(\frac{f(120)-f(60)}{120-60}=\frac{0.1-0.1}{60}=0\)
By the Mean Value Theorem, there must exist a c, for 60< c < 120, such that f ′( c) = 0.
1(c) The rate of flow of gasoline, in gallons per second, can also be modeled by \(g(t)=\left ( \frac{t}{500} \right )cos\left ( (\frac{t}{120})^{2} \right )\) for 0 ≤ t ≤150. Using this model, find the average rate of flow of gasoline over the time interval 0 ≤ t ≤ 150. Show the setup for your calculations.
\(\frac{1}{150-0}\int_{0}^{150}g(t)dt\)
= 0.0959967
1(d) Using the model g defined in part (c), find the value of g′(140 .) Interpret the meaning of your answer in the context of the problem.
g′(140) ≈ −0.004908
g′(140) = − 0.005 (or −0.004 )
The rate at which gasoline is flowing into the tank is decreasing at a rate of 0.005 (or 0.004 ) gallon per second per second at time t = 140 seconds