Calculus derivatives word problem

In summary, to determine the speed of passing cars while standing in the front hall of the school, one must measure the distance from the building to the road, the distance from the window to where you are standing, and the width of the window. Then, using the formula speed = distance/time, the time it takes for the car to pass in front of the window can be divided by the distance calculated from the measurements to determine the speed. This method does not require measuring angles.
  • #1
robin123
10
0

Homework Statement


Is it possible to accurately approximate the speed of a passing car while standing in the
protected front hall of the school?
Task: Determine how fast cars are passing the front of the school. You may only go
outside to measure the distance from where you are standing to the road. All
other measurements must be taken from inside the building (I want you to stay
warm and dry!). Make sure to clearly explain your process to gauge the speed of
passing cars and submit all of your work!

Homework Equations


speed= distance/time

The Attempt at a Solution


time can be calculated for distance we have to suppose an angle and working with triangles but i am not able to figure out how

<Moderator's note: Approved as OP doesn't ask for a solution but just a guidance.>
 
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  • #2
robin123 said:

Homework Statement


Is it possible to accurately approximate the speed of a passing car while standing in the
protected front hall of the school?
Task: Determine how fast cars are passing the front of the school. You may only go
outside to measure the distance from where you are standing to the road. All
other measurements must be taken from inside the building (I want you to stay
warm and dry!). Make sure to clearly explain your process to gauge the speed of
passing cars and submit all of your work!

Homework Equations


speed= distance/time

The Attempt at a Solution


time can be calculated for distance we have to suppose an angle and working with triangles but i am not able to figure out how

<Moderator's note: Approved as OP doesn't ask for a solution but just a guidance.>

Draw a section of the road and the perpendicular line from where you are standing to the road. That's the distance you know. With a surveyor's transit measure the angles to the starting and ending point where you can see the cars, and draw those two lines. Now use trigonometry and a timer.
 
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  • #3
LCKurtz said:
With a surveyor's sextant measure the angles to the starting and ending point where you can see the cars,
Or even use just your protractor from your Trigonometry class, but it won't be as accurate. For extra credit, estimate your error when using the protractor versus using a super-accurate instrument to measure the angles to the first and last positions of the cars... :smile:
 
  • #4
@robin123 a similar problem, but with one less step (no speed involved) is how would you figure out the height of a flagpole with a protractor, a long tape measure, and no ladder.
 
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  • #5
LCKurtz said:
Draw a section of the road and the perpendicular line from where you are standing to the road. That's the distance you know. With a surveyor's sextant measure the angles to the starting and ending point where you can see the cars, and draw those two lines. Now use trigonometry and a timer.
will there be 2 lines with different angles? does that mean i will have two diagrams?
 
  • #6
robin123 said:
will there be 2 lines with different angles? does that mean i will have two diagrams?
Just show us a sketch of your building, the road, the distance to the road, and the angles to the initial and final car positions as it passes by. Use the Upload button at the lower right of the Reply window to Upload a PDF or JPEG of your sketch.
 
  • #7
berkeman said:
Just show us a sketch of your building, the road, the distance to the road, and the angles to the initial and final car positions as it passes by. Use the Upload button at the lower right of the Reply window to Upload a PDF or JPEG of your sketch.
will it look like this?
 

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  • #8
Yes. You could measure on the other side of the perpendicular too until the car gets out of sight. Do you know how to use trigonometry to figure out how far the car goes while it is in sight and you are timing it?
 
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  • #9
LCKurtz said:
Yes. You could measure on the other side of the perpendicular too until the car gets out of sight. Do you know how to use trigonometry to figure out how far the car goes while it is in sight and you are timing it?
yeah, I can use the trigonometry but for that other side, I need an angle. I am confused about that angle. part. For the time I will start with 0s and till the time I can see it.
 

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  • #10
I would do it with two angles. The angle ##\theta## you have drawn for the car approaching the perpendicular and another angle ##\alpha## on the other side of the perpendicular as the car leaves the perpendicular. Those angles aren't zero. They are what you are going to measure so they are known.
 
  • #11
LCKurtz said:
I would do it with two angles. The angle ##\theta## you have drawn for the car approaching the perpendicular and another angle ##\alpha## on the other side of the perpendicular as the car leaves the perpendicular. Those angles aren't zero. They are what you are going to measure so they are known.
You do not need to measure angles . Measure the distance between the road and the wall, D. Stand at a distance d from the window of your room. Measure the width of the window: w. Measure the time you see the car passing in front of the window. The triangles shown are similar, so you can calculate the distance s the car travels from the measured data: D, d, and w.
upload_2018-5-18_6-45-48.png
 

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  • #12
@ehild: Aren't you assuming the building and window are parallel to the road?
 
  • #13
LCKurtz said:
@ehild: Aren't you assuming the building and window are parallel to the road?
Yes, I assume. It is quite common, that the buildings are parallel to the road isn't it?
Otherwise the problem would be tricky to measure angles. And the distance of the road from the building would not be constant along the route of the car.
 
  • #14
ehild said:
You do not need to measure angles . Measure the distance between the road and the wall, D. Stand at a distance d from the window of your room. Measure the width of the window: w. Measure the time you see the car passing in front of the window. The triangles shown are similar, so you can calculate the distance s the car travels from the measured data: D, d, and w.
View attachment 225873
i am confused that how D,d,w can help us calculate s
 
  • #15
upload_2018-5-18_8-7-26.png

The AOB triangle is similar to POQ triangle. What is the ratio s/w in terms of d and D?
 

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  • #16
ehild said:
View attachment 225877
The AOB triangle is similar to POQ triangle. What is the ratio s/w in terms of d and D?
if the above two triangles are similar, that's mean s should be equal to w?
 
  • #17
robin123 said:
if the above two triangles are similar, that's mean s should be equal to w?
NO. They are not congruent. What does similar mean in geometry?
 
  • #18
ehild said:
NO. They are not congruent. What does similar mean in geometry?
Sides are proportional
 

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  • #19
robin123 said:
Sides are proportional
Which sides are proportional?
 
  • #20
ehild said:
Which sides are proportional?
Is that the way I am going to find D?
 

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  • #21
Sorry, I can not read your writing.
S is the length taken by the car. It is the base of the big triangle.
W is the width of the window, it is the base of the small triangle.
D is the distance of the road from the wall. (Measured)
d is your distance from the wall inside the building (also measured) .
The height of the small triangle is d.
What is the height of the big triangle?
The ratio of heights of the triangles is the same as the ratio of the bases. So what is s/w in terms of d and D?
 
  • #22
ehild said:
Sorry, I can not read your writing.
S is the length taken by the car. It is the base of the big triangle.
W is the width of the window, it is the base of the small triangle.
D is the distance of the road from the wall. (Measured)
d is your distance from the wall inside the building (also measured) .
The height of the small triangle is d.
What is the height of the big triangle?
The ratio of heights of the triangles is the same as the ratio of the bases. So what is s/w in terms of d and D?
in that case, we can find S by s/w=D/d we have three things given and we have to find S
 
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  • #23
So, what formula do you get for the speed of the car using the window method?
Also, since you said you can do the trig, what formula do you get using the one or two angles method mentioned earlier?
 
  • #24
LCKurtz said:
So, what formula do you get for the speed of the car using the window method?
Also, since you said you can do the trig, what formula do you get using the one or two angles method mentioned earlier?
isn't the formula still the same that speed=distance traveled by car / the time it took to travel that distance
if I have the angle I can just use the height and tan(theta) to figure out the distance traveled by car.
s/d=tan(θ)
 
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1. What is a derivative in calculus?

A derivative in calculus is a mathematical concept that represents the rate of change of a function with respect to its independent variable. It is the slope of the tangent line at a specific point on a curve and can be used to find the instantaneous rate of change at that point.

2. How do I solve a calculus derivative word problem?

To solve a calculus derivative word problem, you first need to identify the function and the independent variable. Then, use the rules of differentiation, such as the power rule or chain rule, to find the derivative of the function. Finally, plug in the given values to find the specific rate of change at a certain point.

3. What is the difference between a derivative and a rate of change?

A derivative is a mathematical concept that represents the rate of change of a function at a specific point, while a rate of change is a general term that describes how much a quantity is changing over a given interval. Derivatives are used to find the instantaneous rate of change, while rates of change can be calculated for any given interval.

4. Can derivatives be negative?

Yes, derivatives can be negative. A negative derivative represents a decreasing function, where the slope of the tangent line is negative. This means that the function is decreasing as the independent variable increases.

5. How are derivatives used in real life?

Derivatives are used in many real-life scenarios, such as calculating the speed of an object at a specific time, finding the maximum or minimum value of a function, and determining the rate of change of a quantity in various physical and economic systems. They are also used in engineering, physics, and finance to model and analyze real-world situations.

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