AP Calculus BC 4.5 Solving Related Rates Problems - MCQs - Exam Style Questions
Calc-Ok Question
A thin rectangular sheet of metal expands. At the instant when the width is \(200\) mm and the length is \(300\) mm, the width increases at \(2\) mm/min and the length increases at \(3\) mm/min. What is the rate of change of the area (mm\(^2\)/min) at that instant?
(A) \(5\)
(B) \(6\)
(C) \(600\)
(D) \(1200\)
(B) \(6\)
(C) \(600\)
(D) \(1200\)
▶️ Answer/Explanation
Detailed solution
Area \(A=L \times W\).
Differentiate:
\(\displaystyle \frac{dA}{dt}=L \times \frac{dW}{dt}+W \times \frac{dL}{dt}.\)
Substitute (mm, min):
\(L=300,\quad W=200,\quad \dfrac{dW}{dt}=2,\quad \dfrac{dL}{dt}=3.\)
Compute:
\(\displaystyle \frac{dA}{dt}=300 \times 2+200 \times 3=600+600=1200\ \text{mm}^2/\text{min}.\)
✅ Answer: (D)
No-Calc Question
Paint spills onto a floor in a circular pattern. The radius of the spill increases at a constant rate of \(2.5\) inches per minute. How fast is the area of the spill increasing when the radius of the spill is \(18\) inches?
(A) \(5\pi\ \text{in}^2/\text{min}\)
(B) \(36\pi\ \text{in}^2/\text{min}\)
(C) \(45\pi\ \text{in}^2/\text{min}\)
(D) \(90\pi\ \text{in}^2/\text{min}\)
(B) \(36\pi\ \text{in}^2/\text{min}\)
(C) \(45\pi\ \text{in}^2/\text{min}\)
(D) \(90\pi\ \text{in}^2/\text{min}\)
▶️ Answer/Explanation
Detailed solution
Area of a circle: \(A=\pi r^2\).
Differentiate: \(\displaystyle \frac{dA}{dt}=2\pi r\,\frac{dr}{dt}\).
At \(r=18\) and \(\dfrac{dr}{dt}=2.5\):
\(\displaystyle \frac{dA}{dt}=2\pi\times 18\times 2.5=90\pi.\)
✅ Answer: (D)