Question

Find the probability that a dart hits one of the shaded areas. Thewhite figure is a rectangle. Be sure to show all work.

Answer 1

Given the question in the image, the following are the solution steps to answer the question.

STEP 1: Get the angles of the hexagon

The internal angles of an hexagon is given as:

`\begin{gathered} (180(n-2))/(n) \\ n=6\text{ since hexagon has 6 sides} \\ So\text{ we have:} \\ (180(6-2))/(6)=(180(4))/(6)=(720)/(6)=120\degree \end{gathered}`

Therefore each angle of the hexagon is 120 degrees.

STEP 2: find the length of the sides

We remove the right triangles as seen below:

Using the special right triangles, we have:

STEP 3: find the area of the extracted triangle above

`\begin{gathered} b=1,h=√(3) \\ Area=(1)/(2)\cdot1\cdot√(3)=(√(3))/(2)units^2 \end{gathered}`

Since there are two right triangles, we multiply the area by 2 to have:

`Area=2\cdot(√(3))/(2)=√(3)`

There are two triangles(both sides), therefore the total area of the shaded area will be:

`√(3)\cdot2=2√(3)`

STEP 4: Find the area of the whole hexagon

`\begin{gathered} Area=(3√(3)s^2)/(2) \\ s=hypotenuse\text{ of the right triangle}=2 \\ Area=(3√(3)\cdot4)/(2)=6√(3) \end{gathered}`

STEP 5: Find the probability

`\begin{gathered} Probability=\frac{possible\text{ area}}{Total\text{ area}} \\ \\ Possible\text{ area}=2√(3) \\ Total\text{ area}=6√(3) \\ \\ Probability=(2√(3))/(6√(3))=(1)/(3)=0.3333 \end{gathered}`

Hence, the probability that the dart hits one of the shaded areas is approximately 0.3333