Question

1. probability of shaded area: ________

Explanation:

Step1: Calculate the area of the rectangle

The area formula for a rectangle is $A = l\times w$. Here, $l = 8$ in and $w = 6$ in. So $A_{rectangle}=8\times6 = 48$ in².

Step2: Calculate the diameter of the circle

The diagonal of the rectangle is the diameter of the circle. Using the Pythagorean theorem $d=\sqrt{6^{2}+8^{2}}=\sqrt{36 + 64}=\sqrt{100}=10$ in, then the radius $r=\frac{d}{2}=5$ in.

Step3: Calculate the area of the circle

The area formula for a circle is $A=\pi r^{2}$. So $A_{circle}=\pi\times5^{2}=25\pi$ in².

Step4: Calculate the probability

The probability $P$ of a point being in the shaded - rectangle area is the ratio of the area of the rectangle to the area of the circle. $P=\frac{A_{rectangle}}{A_{circle}}=\frac{48}{25\pi}$.

Answer:

$\frac{48}{25\pi}$