AP Calculus AB 8.3 Using Accumulation Functions and Definite Integrals in Applied Contexts - MCQs - Exam Style Questions
Calc-Ok Question
A metal rod has temperature \(75^\circ\!F\) at \(t=0\). The temperature is increasing at a rate \(47e^{-0.2t}\) degrees Fahrenheit per minute for \(0\le t\le 10\). To the nearest degree, what is the temperature at \(t=10\) minutes?
(A) \(81^\circ\!F\)
(B) \(203^\circ\!F\)
(C) \(278^\circ\!F\)
(D) \(545^\circ\!F\)
(B) \(203^\circ\!F\)
(C) \(278^\circ\!F\)
(D) \(545^\circ\!F\)
▶️ Answer/Explanation
Temperature change \(=\displaystyle \int_{0}^{10} 47e^{-0.2t}\,dt =47\left[-\frac{1}{0.2}e^{-0.2t}\right]_{0}^{10} =235\,(1-e^{-2})\approx 203.2.\)
Final temperature \(=75+203.2\approx 278^\circ\!F\).
✅ Answer: (C)
Final temperature \(=75+203.2\approx 278^\circ\!F\).
✅ Answer: (C)
Calc-Ok Question
The rate of change of the water level in a lake at time \(t\) months is \(r(t)\) feet per month. The graph of \(r\) is shown for \(0\le t\le 12\). For which times \(t\) with \(0<t\le 12\) is the water level equal to its level at \(t=0\)?
(A) \(t=5\) only
(B) \(t=5\) and \(t=12\)
(C) \(t=2,\ t=5,\) and \(t=8\)
(D) \(t=1\) and \(t=6\)
(B) \(t=5\) and \(t=12\)
(C) \(t=2,\ t=5,\) and \(t=8\)
(D) \(t=1\) and \(t=6\)
▶️ Answer/Explanation
Water-level change from \(0\) to \(t\) is \(\displaystyle \int_{0}^{t} r(x)\,dx\).
From the graph, the signed area on \([0,2]\) is negative and has magnitude \(3\).
The area on \([2,5]\) is positive with magnitude \(3\) → net change \(0\) at \(t=5\).
The segment \([5,8]\) contributes \(-3\) , and \([8,12]\) adds \(+3\), which cancels the \(-3\) → net change \(0\) again at \(t=12\).
✅ Answer: (B)
From the graph, the signed area on \([0,2]\) is negative and has magnitude \(3\).
The area on \([2,5]\) is positive with magnitude \(3\) → net change \(0\) at \(t=5\).
The segment \([5,8]\) contributes \(-3\) , and \([8,12]\) adds \(+3\), which cancels the \(-3\) → net change \(0\) again at \(t=12\).
✅ Answer: (B)