MA005 Calculus 1 Coaching Session 1.2.1 # 5

ma005

#1

Hi this is Swapnila! I am a Student Success Coach and will be explaining some practice problems from the MA005 Calculus course. Today we will review an assigned problem from Unit 1. In the “Homework Assessment” you were asked to solve this problem:

image

a) We will use following approach to solve this part:

  1. First find the distance traveled by the airplane at the given time.
  2. Denote the angle at intersection of direction of airplane and telescope as θ
  3. Plug the values of opposite and adjacent sides in the slope formula.

Step 1: First find the distance traveled by the airplane at the given time:
Distance = velocity ∙ time
i) After t = 5 seconds.
Distance = 300 ∙ 5 = 1500 feet
ii) After t =10 seconds.
Distance = 300∙10 = 3000 feet
iii) After t = 20 seconds.
Distance = 300 ∙ 20 = 6000 feet

Step 2: Denote the angle at intersection of direction of airplane and telescope as θ:
image

Step 3: Plug the values of opposite and adjacent sides in the slope formula:
Slope formula:
image
i) After t=5 seconds:
The distance traveled by the airplane is 1500 feet, which is the adjacent side, and altitude of airplane is 5000 feet, which is the opposite side, as shown in the figure below:

image

Plug these values in the formula:
image
Simplify:
image

ii) After t=10 seconds:
The distance traveled by the airplane is 3000 feet, which is the adjacent side, and altitude of airplane is 5000 feet, which is the opposite side, as shown in the figure below:
image
Plug these values in the formula:
image
Simplify:
image

iii) After t=20 seconds:
The distance traveled by the airplane is 6000 feet, which is the adjacent side, and altitude of airplane is 5000 feet, which is the opposite side, as shown in the figure below:
image
Plug these values in the formula:
image
Simplify:
image

Conclusion:
By using slope formula,
i) The slope of the telescope after 5 seconds is image
ii) The slope of the telescope after 10 seconds is image
iii) The slope of the telescope after 20 seconds is image

b) To find the slope of the telescope t seconds after the plane passes overhead, use the slope formula. Our approach for this problem will be as follows:

  1. First find the distance traveled by the airplane at the given time t.
  2. Denote the angle at intersection of direction of airplane and telescope as θ.
  3. Plug the values of opposite and adjacent side in the slope formula.

Step 1: First find the distance traveled by the airplane at the given time:
Distance = velocity ∙ time
At t seconds,
Distance = 300 ∙ t = 300 t feet
Step 2: Denote the angle at intersection of direction of airplane and telescope as θ:
image
Step 3: Plug the values of opposite and adjacent side in the slope formula.
Slope formula:
image
At t seconds:
The distance traveled by the airplane is 300t feet, which is the adjacent side, and the altitude of the airplane is 5000 feet, which is the opposite side, as shown in the figure below:
image
Plug these values in formula:
image
Simplify:
image

Conclusion:
By using slope formula,
The slope of the telescope after t seconds is image,t > 0

c) The approach for this part will be as follows:

  1. Use the result from part b, and check how the slope is changing as t increases.

Step 1: Use the result from part b, and check how the slope is changing as t increases.
The slope of the telescope after t seconds is image,t > 0.
Here, the numerator remains constant, and the denominator depends on t. As t increases, the denominator increases, and so, the result of the slope decreases.

Conclusion:
After it passes overhead, the slope of the telescope is decreasing.

In addition to this, I am also attaching the pdf document of the practice problem for your ready reference.
Question 5.pdf (309.7 KB)

Please let me know if you have any question on this problem, or on this topic generally. I will be here in the forum for the next hour.