ShmoopTube

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Here’s your shmoop du jour, brought to you by lawn darts.

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Because there’s just about nothing safer and more fun than hurling sharp projectiles across the back yard.

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If Caitlin throws a lawn dart so that it lands inside the trapezoid, what is the approximate

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probability that it will land inside the circle?

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Here are the potential answers...

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Okay, we’ve got this gray area – the circle, and…everything else.

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We want to know the area of that circle compared to the area that isn’t inside the circle.

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The easy peasy way is to figure out the area of the trapezoid,

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figure out the area of the circle, and then divide the latter by the former.

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Let’s start with the circle, because it looks… well, easier.

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The formula for the area of a circle is pi r squared.

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We know the circle has a diameter of 8, making the radius 4.

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So… pi times 4 squared, or pi times 16… is roughly 50.24.

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Moving on. Trapezoid time.

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The area of a trapezoid is slightly more complicated…

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One-half of base 1 plus base 2…times height.

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Our top base is 8…

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...but it seems the problem doesn't want to be as generous when it comes to the bottom base.

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What if we make that right portion of the trapezoid into a triangle?

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We get a triangle with a side of 8 and a hypotenuse of 10.

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Well, hey – we recognize that sucker! It’s a classic 3-4-5 triangle…

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meaning that the legs and hypotenuse have measurements that are multiples of 3, 4 and 5.

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8 is double 4 and 10 is double 5…

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...we have to double 3 to get the length of the remaining leg.

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Which is 6.

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So this leg is 6…plus 8…gives us a grand total of 14 for the bottom base.

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Now we can plug into our trapezoid formula:

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1/2 times 8 plus 14, or 22, times our height of 8…gives us 88.

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Now don’t forget we have to divide the area of the circle by the area of the trapezoid.

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So we have 50.24 divided by 88 gives us right around 57 percent… answer D.

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All right, crisis averted.

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Caitlin has been captured and strapped down.

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We may now all enjoy the rest of our picnic.

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