Answer:
About 68.2%.
Step-by-step explanation:
We want to find the probability that a randomly selected point lands in the white area.
To do so, we can find the probability that the point does not land in the white area (i.e. the red area) and then subtract it from 100%.
The circle has a radius of 4 cm. Note that the base of the triangle is the diamater, so the base measures 8 cm. The height of the triangle is the radius, so it measures 4 cm. Thus, the area of the triangle is:
[tex]\displaystyle \begin{aligned} A &= \frac{1}{2}bh \\ &= \frac{1}{2}(8)(4) \\ &= 16\text{ cm}^2\end{aligned}[/tex]
Likewise, the area of the entire circle will be:
[tex]\displaystyle \begin{aligned} A &= \pi r^2 \\ &= \pi(4)^2 \\ &=16\pi \text{ cm}^2\end{aligned}[/tex]
Then the probability that a randomly seleted point lands in the red area is:
[tex]\displaystyle P\left(\text{Red}\right) = \frac{\text{Red}}{\text{Total}} =\frac{16}{16\pi} = \frac{1}{\pi}[/tex]
The probability that the point lands in the red area or the non-red area (i.e. white area) must total 100%. In other words:
[tex]\displaystyle P\left(\text{Red}\right) + P\left(\text{White}\right) = 1[/tex]
Thus:
[tex]\displaystyle P\left(\text{White}\right) = 1 - P\left(\text{Red}\right)[/tex]
Substitute:
[tex]\displaystyle P\left(\text{White}\right) = 1 - \left(\frac{1}{\pi}\right)[/tex]
Use a calculator. Hence:
[tex]\displaystyle P\left(\text{White}\right) \approx 0.6817 = 68.2\%[/tex]
The probability that a randomly selected point lands in the white area is about 68.2%.
