Sustainable Gardening Tips for Beginners

In summary, the magnification of an object is the same as the image distance over the object distance.
  • #1
DaalChawal
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1628661425748.png
 
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  • #2
Hi DaalChawal,

First a diagram:
\begin{tikzpicture}[scale=2]
\def\u{2.5}
\def\v{3.75}
\def\x{1.5}
\def\f{1.5}
\coordinate[label=above: A] (A) at ({-(\u+2)},1.5);
\coordinate[label=below: D] (D) at ({-(\u+2)},-1.5);
\coordinate[label=above: B] (B) at (-\u,.5);
\coordinate[label=below: C] (C) at (-\u,-.5);
\coordinate[label=F] (F) at (-\f,0);
\coordinate[label=F] (F') at (\f,0);

\coordinate[label=below: A'] (A') at ({\v-\x},{-\x/2});
\coordinate[label=above: D'] (D') at ({\v-\x},{\x/2});
\coordinate[label=below: B'] (B') at (\v,{-\x/2});
\coordinate[label=above: C'] (C') at (\v,{\x/2});

\draw (-6,0) -- (5,0);
\draw[ultra thick] (0,-2) -- (0,2) node[ above ] {+};
\filldraw (F) circle (.03);
\filldraw (F') circle (.03);
\draw (A) -- (B) -- (C) -- (D) -- cycle;
\draw (A') -- (B') -- (C') -- (D') -- cycle;
\draw[yshift=-0.1cm, latex-latex] ({-(\u+2)},0) -- node[ below ] {2} (-\u,0);
\draw[xshift=-0.1cm, latex-latex] ({-(\u+2)},1.5) -- node[ above left] {3} ({-(\u+2)},-1.5);
\draw[xshift=-0.1cm, latex-latex] (-\u,0.5) -- node[ above left] {1} (-\u,-0.5);
\draw[yshift=-0.1cm, latex-latex] ({\v-\x)},{\x/2}) -- node[ below ] {$x$} (\v,{\x/2});
\draw[xshift=0.1cm, latex-latex] (\v,{\x/2}) -- node[ above right ] {$x$} (\v,{-\x/2});
\draw[yshift=-0.1cm, latex-latex] (-\u,0) -- node[ below ] {$u$} (0,0);
\draw[yshift=-0.1cm, latex-latex] (0,0) -- node[ below ] {$v$} (\v,0);

\draw[help lines] (B) -- (B');
\draw[help lines] (B) -- (0,0.5) -- (B');
\draw[help lines] (B) -- (0,{-\f/(\u-\f)*0.5}) -- (B');
\draw[help lines] (A) -- (A');
\draw[help lines] (A) -- (0,1.5) -- (A');
\draw[help lines] (A) -- (0,{-\f/((\u+2)-\f)*1.5}) -- (A');

\end{tikzpicture}Let $x$ be the side length of the square in the image.
Let $f$ be the focal length of the lens.
Let $u$ be the distance of $BC$ to the lens and let $v$ be the distance of its image to the lens.
Then $u+2$ is the distance of $AD$ to the lens, and $v-x$ is the distance of its image to the lens.

The magnification of an object, which is the size of the image over the size of the object, is the same as the image distance over the object distance.
So the magnification of $BC$ is: $\frac{B'C'}{BC}=\frac x 1=\frac{v}{u}$.
The magnification of $AD$ is: $\frac{A'D'}{AD} = \frac x 3 = \frac{v-x}{u+2}$.
Furthermore, the lens formula tells us that $\frac 1u+\frac 1v=\frac 1f$ and $\frac 1{u+2}+\frac 1{v-x}=\frac 1f$.

So:
\begin{cases}\frac x 1=\frac{v}{u} \\ \frac x 3= \frac{v-x}{u+2} \\ \frac 1u+\frac 1v=\frac 1f \\ \frac 1{u+2}+\frac 1{v-x}=\frac 1f
\end{cases}

Solve?
 
Last edited:
  • #3
Thanks, I got it now.
 
  • #4
I had actually made a couple of mistakes.
The image should be upside down .
And the image of $AD$ is closer instead of further away. That is, its image distance is $v-x$ instead of $v+x$.
It becomes apparent when we actually try to solve the equations.

I've updated the drawing and the formulas in my previous post.
And I've also added some extra help lines to show that it actually works and is to scale.
 
  • #5
Well sir, I only read your solution that what was the concept used and then I tried on my own and I got the answer 😅
 

Related to Sustainable Gardening Tips for Beginners

1. What is sustainable gardening?

Sustainable gardening involves using practices that are environmentally friendly and promote long-term health and productivity of the garden. This includes using organic methods, conserving resources, and minimizing waste.

2. Why is sustainable gardening important?

Sustainable gardening helps to protect and preserve the environment by reducing the use of harmful chemicals and promoting biodiversity. It also helps to conserve resources such as water and energy, and can save money in the long run.

3. How can I start practicing sustainable gardening?

Some simple ways to start practicing sustainable gardening include using organic fertilizers and pest control methods, composting, conserving water, and choosing native plants that are well-suited to your climate and soil.

4. What are some common mistakes to avoid in sustainable gardening?

Some common mistakes to avoid include over-fertilizing, using non-biodegradable materials, and not properly managing pests and weeds. It's also important to avoid using invasive plants and to properly dispose of any hazardous materials.

5. Can I still have a beautiful garden while practicing sustainable gardening?

Absolutely! Sustainable gardening does not mean sacrificing aesthetics. In fact, by choosing native plants and incorporating natural elements like rocks and water features, you can create a beautiful and sustainable garden that is also beneficial for the environment.

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