Solve Dxdy Word Problem: Height of Bulb at

In summary, we can use geometry and the equation of a circle to find the height of the bulb in the figure. We can start by setting up a right triangle and using similarity to find the height. Another approach is to use the equation of the line tangent to the circle at the given point and solving for the height. This can be done by substituting the coordinates of the point into the equation of the tangent line and using the equation of the circle to find the corresponding y-coordinate. Finally, we can also use the general fact that a tangent to a circle at a given point can be expressed as a linear equation.
  • #1
karush
Gold Member
MHB
3,269
5
View attachment 4671

how high h must the bulb in the figure be if point
is right on the edge of the illuminated region.






hopefully if the is setup ok what is next?
 
Last edited:
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  • #2
The equation for the circle is incorrect.
 
  • #3
I copied the figure wrong so OP has pic from book
 
  • #4
This can be solved using geometry. Let be the origin, be the point of tangency of the sloped line and the circle, and let have coordinates . Then the right triangle is similar to the large right triangle with side . From there can be found.

What do you mean by dxdy in the thread title?
 
  • #5






When Then
 
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  • #6
I would likely work this as follows:

The equation of the line is given by:



Now, plug this into the equation of the circle:







Require the discriminant to be zero since the line and circle are tangent:



This reduces to:



Discarding the negative root, we are left with:

 
  • #7
Well that was very helpful.

It was hard to see how to use the circle in this.
 
  • #8
We can use still another general fact that a tangent to a circle at point is

(More generally, the tangent to at point is .) Substituting and into (*) we find and from it follows that . Thus, the tangent equation is , from where when .
 

FAQ: Solve Dxdy Word Problem: Height of Bulb at

What does "Dxdy" mean in the word problem?

"Dxdy" in this word problem refers to the derivative of the function that represents the height of the bulb over time.

How do I solve for the height of the bulb at a specific point?

To solve for the height of the bulb at a specific point, you will need to use the formula for the derivative of a function. This will give you the rate of change of the height at that point, and you can then use that rate to find the actual height by integrating over a certain time interval.

Can I use any point on the graph to find the height of the bulb?

Yes, you can use any point on the graph to find the height of the bulb as long as you have the derivative of the function at that point. This will give you the rate of change, which you can then use to calculate the height.

What does the point (1.25, 0) represent in this word problem?

The point (1.25, 0) represents a specific time and height on the graph. The x-coordinate of 1.25 represents the time in seconds, and the y-coordinate of 0 represents the height in meters.

Can I solve this word problem without knowing the derivative of the function?

No, in order to solve this word problem, you will need to know the derivative of the function at the point (1.25, 0). Without this information, it is not possible to accurately calculate the height of the bulb at that point.

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