Find the radius!

Geometry Level 5

On a field of grass, a lamb is tied to a pole with a 1 meter long rope. After the lamb has eaten all the available grass in the form of a circle, the pole is moved on the circumference of the first circle. How many meters does the new rope need to be, so the lamb can eat the same amount of grass?

Give the solution up to 2 decimals (calculator allowed).

Hint:

The answer is 1.25.

3 solutions

Chew-Seong Cheong
Mar 30, 2020

I offer another hint. We note that the total grass area grazed by the lamb is $2\pi \text{ m}^2$ . This area is equal to the sum of the area of sector $OPR$ , area of sector $QPR$ (where $r$ is the length of the rope) and area of quadrilateral $OPQR$ . In equation, we have:

$\begin{aligned} A_{OPR} + A_{QPR} + A_{OPQR} & = 2\pi \\ (\pi - \blue \theta )(1^2)+ \left(\frac {\pi + \blue \theta}2\right)r^2 + \red{2\left(\frac r2\sqrt{1-\frac {r^2}4}\right)} & = 2\pi & \small \blue{\text{Note that } \theta = \angle POQ = \angle ROQ= 2 \sin^{-1} \frac r2} \\ 2(r^2-2)\sin^{-1} \frac r2 + \pi r^2 + r\sqrt{4- r^2} & = 2 \pi & \small \red{\text{Area of 2 triangles}} \end{aligned}$

Solving the equation numerically, we get $r \approx \boxed{1.25}$ .

great solution, thanks for adding it.

Sam sam - 1 year, 2 months ago

Glad that you like it. It should be a Geometry problem instead of Calculus one.

Chew-Seong Cheong - 1 year, 2 months ago

I'm struggling to see why the sum you've written equals to $2 \pi$ .

I see why the grand total area is $2 \pi \, m^2$ , but why would that sum also be equal to $2 \pi$ ?

Fernando Tsurukawa - 1 year, 2 months ago

I have recolored the figure. Hope that it helps.

Chew-Seong Cheong - 1 year, 2 months ago

Ohh, I see it now! Thank you! I'm changing my reaction emoji to "Brilliant"!

Fernando Tsurukawa - 1 year, 2 months ago

The question framing is not correct I think because according to that the lamb should have eaten all the grass covering area of circle with centre O

Srinjoy G - 1 year, 2 months ago

The question says, first pole or the center of the circle is at $(0,0)$ and the radius is 1 m. Then the pole is moved to $(1.0)$ as shown in the figure above.

Chew-Seong Cheong - 1 year, 2 months ago

The lamb has eaten all the available grass in the form of a circle means a lamb has eaten all the available grass in the whole circle the area of which is $\pi*1m^2$ .

Winod DHAMNEKAR - 1 year, 2 months ago

Niranjan Khanderia - 1 year, 2 months ago

Can you give me solution for the last eq.?

Dev Dwivedi - 1 year, 1 month ago

I used numerical method to solve it. Some problems may not be able to be solve algebraically. I used an Excel spreadsheet to estimate the value $r$ .

Chew-Seong Cheong - 1 year, 1 month ago

Sam Sam
Mar 30, 2020

Vinod Kumar
May 1, 2020

plot acos[(2-r^2)/2]+r^2[acos(r/2)]-(1/2) r √(4-r^2)+π-π(r^2) for 1.252>r>1.251

Find the radius!

The answer is 1.25.

3 solutions

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