A dart is thrown and hits the square target shown. The target has a circle inscribed in it, and within that circle, is inscribed another square. If the dart is equally likely to hit any point on the target, what is the probability that the dart will land on the shaded area? Not sure how I'm supposed to go about this one...

Question

yoyo2443 · Answer

The graph:

jim_thompson5910 · Answer

Are you able to determine the radius of the circle?

yoyo2443 · Answer

I'm  guessing the radius is $$\sqrt{2}$$

yoyo2443 · Answer

From what I see on the image anyways.

jim_thompson5910 · Answer

yes

jim_thompson5910 · Answer

what is the side length of the smaller square?

yoyo2443 · Answer

Not sure...

jim_thompson5910 · Answer

do you see how the side that has label "1" is half of the side length of the smaller square?

yoyo2443 · Answer

Oooh, I see it now, yeah!

jim_thompson5910 · Answer

shaded area = (area of circle) - (area of smaller square)

are you able to compute the area of the shaded region?

yoyo2443 · Answer

One moment let me write it out on paper.

yoyo2443 · Answer

$$A=\pi \sqrt{2}^2$$ $$A=\pi 2$$ $$A=6.28$$ Is what I got

jim_thompson5910 · Answer

yes 2pi or approx 6.28

jim_thompson5910 · Answer

that's the area of the circle

jim_thompson5910 · Answer

what's the area of the small square?

yoyo2443 · Answer

$$A=2^2$$ which would make it 4.

jim_thompson5910 · Answer

yes

jim_thompson5910 · Answer

shaded area = (area of circle) - (area of smaller square)

shaded area = 2pi - 4 ... exact area

shaded area = 6.28 - 4

shaded area = 2.28     .... approx area

jim_thompson5910 · Answer

agreed?

yoyo2443 · Answer

Yep!

jim_thompson5910 · Answer

now calculate the area of the larger square

yoyo2443 · Answer

Assuming the side length is about 4..? making it 16..?

robtobey2 · Answer

$$\frac{\pi  \sqrt{2}^2-2^2}{\left(2 \sqrt{2}\right)^2}=\frac{1}{8} (2 \pi -4)=\frac{\pi -2}{4}=0.285398 $$

jim_thompson5910 · Answer

`Assuming the side length is about 4..? making it 16..? `
incorrect @yoyo2443

jim_thompson5910 · Answer

sqrt(2) makes up half the side length of the bigger square

yoyo2443 · Answer

so.. $$\sqrt{2}+\sqrt{2}=2.82$$ ?

jim_thompson5910 · Answer

or 2*sqrt(2)

jim_thompson5910 · Answer

square 2*sqrt(2) to get what?

yoyo2443 · Answer

To get the side length of the bigger square, 2.82

jim_thompson5910 · Answer

[2*sqrt(2)]^2 = 2^2*[sqrt(2)]^2

[2*sqrt(2)]^2 = 4*2

[2*sqrt(2)]^2 = 8

so the area of the bigger square is 8

jim_thompson5910 · Answer

probability of hitting shaded region = (area of shaded region)/(area of bigger square)

yoyo2443 · Answer

Ahhh okay so $$Probability= (2.28)(8)=18.24$$

jim_thompson5910 · Answer

you divide, not multiply

yoyo2443 · Answer

Oh woops, I missed the '/ '$$Probability= \frac{ 2.28 }{ 8 }=0.285$$

jim_thompson5910 · Answer

yes that's the approximate probability

jim_thompson5910 · Answer

the exact probability is (2pi-4)/8 = (pi-2)/4

jim_thompson5910 · Answer

using a calculator, 

(pi-2)/4 = 0.28539816339744

so 0.28539816339744 is the more accurate version of the approximate answer

yoyo2443 · Answer

But usually rounded down to 0.28, yeah, I got it now. Thank you very much for the help!

jim_thompson5910 · Answer

no problem