How can I find "dx" in a straight line of electric charge?(Picture)

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The discussion revolves around understanding the relationship between dx and dteta in the context of a straight line of electric charge. The user initially struggles with the equation dx=R/cos^2(teta) dteta and seeks clarification on its derivation. After engaging with others, the user realizes their previous misconceptions about the topic. Ultimately, they express gratitude for the assistance received, indicating a resolution to their confusion. The conversation highlights the importance of grasping the underlying concepts in physics equations.
MatinSAR
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Homework Statement
How can I find "dx" in a straight line of electric charge?(Details in Picture)
Relevant Equations
r=R/cos(teta)
x=rtan(teta)
1646478251956.png

This is the question ... I have it's solution ...
1646478322100.png

1646478364142.png
My problem : I can't understand why dx=R/cos^2(teta) dteta
I have thought many hours but I couldn't find it's reason ... Can anyone please help with this ?!
 
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If ##x = R\tan{\theta}##, then ##dx/d\theta= \dots##
 
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Likes vanhees71 and MatinSAR
ergospherical said:
If ##x = R\tan{\theta}##, then ##dx/d\theta= \dots##
I got it ... I have learned this in a wrong way but now I know what was wrong ... Thanks a lot 🌷
 
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Likes vanhees71 and jim mcnamara
The book claims the answer is that all the magnitudes are the same because "the gravitational force on the penguin is the same". I'm having trouble understanding this. I thought the buoyant force was equal to the weight of the fluid displaced. Weight depends on mass which depends on density. Therefore, due to the differing densities the buoyant force will be different in each case? Is this incorrect?

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