Home / AP Calculus AB: 5.10 Introduction to Optimization Problems – Exam Style questions with Answer- MCQ

AP Calculus AB: 5.10 Introduction to Optimization Problems – Exam Style questions with Answer- MCQ

AP Calculus AB-Questions and Answers - All Topics
Question
Puppy weighs 2.0 lbs at birth, 3.5 lbs at 2 months. Weight grows proportional to itself in first 6 months. Weight at 3 months?
A) 4.2 lbs
B) 4.6 lbs
C) 4.8 lbs
D) 5.6 lbs
E) 6.5 lbs
▶️ Answer/Explanation
Solution
Proportional growth: \( \frac{dw}{dt} = kw \). Solution: \( w(t) = 2e^{kt} \).
Given: \( t=0 \), \( w=2 \); \( t=2 \), \( w=3.5 \).
\( 3.5 = 2e^{2k} \), \( e^{2k} = 1.75 \), \( k = \frac{\ln(1.75)}{2} \).
Weight: \( w(t) = 2(1.75)^{\frac{t}{2}} \).
At \( t=3 \): \( w(3) = 2(1.75)^{1.5} \approx 4.63 \).
Rounded: \( 4.6 \) lbs.
✅ Answer: B) 4.6 lbs.

Question

The flow of oil, in barrels per hour, through a pipeline on July 9 is given by the graph shown above. Of the following, which best approximates the total number of barrels of oil that passed through the pipeline that day?

(A) 500                          (B) 600                      (C) 2,400                         (D) 3,000                      (E) 4,800

▶️Answer/Explanation

Ans:D

Let r(t) be the rate of oil flow as given by the graph, where t is measured in hours. The total r(t)dt. This can be approximated by counting the squares number of barrels is given by \(\int_{0}^{24}r(t)dt\) below the curve and above the horizontal axis. There are approximately five squares with area 600 barrels. Thus the total is about 3,000 barrels.

Question

A railroad track and a road cross at right angles. An observer stands on the road 70 meters south of the crossing and watches an eastbound train traveling at 60 meters per second. At how many
meters per second is the train moving away from the observer 4 seconds after it passes through the intersection?
(A) 57.60                      (B) 57.88                                                   (C) 59.20                                             (D) 60.00                                             (E) 67.40

▶️Answer/Explanation

Ans:A

Question

The position of an object attached to a spring is given by:

\[ y(t) = \frac{1}{6}\cos(5t) – \frac{1}{4}\sin(5t) \]

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:

\[ v(t) = y'(t) = -\frac{5}{6}\sin(5t) – \frac{5}{4}\cos(5t) \]

2. Set velocity equal to zero and solve:

\[ -\frac{5}{6}\sin(5t) – \frac{5}{4}\cos(5t) = 0 \] \[ \Rightarrow \tan(5t) = -\frac{3}{2} \]

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.

Question

Consider all right circular cylinders for which the sum of the height and circumference is 30 centimeters. What is the radius of the one with maximum volume?

(A) 3 cm                        (B) 10 cm                          (C) 20 cm                                     (D)      \(\frac{30}{\pi ^{2}}\)                        (E) \(\frac{10}{π}cm\)

▶️Answer/Explanation

Ans:E

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