AP Calculus AB 5.10 Introduction to Optimization Problems - MCQs - Exam Style Questions
No-Calc Question
Consider a triangle in the \(xy\)-plane with base endpoints \((1,0)\) and \((5,0)\) and the third vertex on \(y=\ln(2x)-\dfrac{1}{2}x+5\) for \(\dfrac{1}{2}\le x\le 8\). What is the maximum area of such a triangle?
(A) \(\dfrac{19}{2}\)
(B) \(2\ln 2+9\)
(C) \(2\ln 4+8\)
(D) \(2\ln 16+2\)
▶️ Answer/Explanation
Base \(=4\). Height \(=y(x)=\ln(2x)-\tfrac{1}{2}x+5\).
\(A(x)=\tfrac{1}{2}\times 4\times y(x)=2\ln(2x)-x+10\).
\(A'(x)=\tfrac{2}{x}-1=0\Rightarrow x=2\).
\(A_{\max}=2\ln 4-2+10=2\ln 4+8\).
✅ Answer: (C) \(2\ln 4+8\)
Question
The position of an object attached to a spring is given by:
where t is time in seconds. In the first 4 seconds, how many times is the velocity of the object equal to 0?
A) Zero
B) Three
C) Five
D) Six
E) Seven
▶️ Answer/Explanation
Solution
To find when velocity equals zero:
1. First find the velocity function by differentiating position:
2. Set velocity equal to zero and solve:
3. Find solutions in [0,4]:
- The period of the trigonometric functions is \( \frac{2π}{5} \approx 1.26 \) seconds
- In 4 seconds, there are \( \frac{4}{2π/5} \approx 3.18 \) periods
- Each period has 2 zero-velocity points (max/min positions)
- Total zero-velocity points: 3 full periods × 2 = 6 points
4. Graphical interpretation:
- The position function \( y(t) \) is a sinusoidal wave
- Velocity is zero at each peak and trough (max/min displacement)
- 6 such points occur in the first 4 seconds
✅ Therefore, the correct answer is D) Six.