To find the probability that a random point placed in a right trapezoid (with an inscribed circle) is not inside the circle, we need to calculate the area of the trapezoid and the area of the circle.
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Calculate the area of the trapezoid: The area \( A \) of a trapezoid can be calculated using the formula:
\[ A = \frac{1}{2} \times (b_1 + b_2) \times h \]
where \( b_1 \) and \( b_2 \) are the lengths of the two bases and \( h \) is the height. For this trapezoid:
- \( b_1 = 12 \) inches (short base)
- \( b_2 = 23 \) inches (long base)
- \( h = 5 \) inches (height)
Plugging in these values:
\[ A = \frac{1}{2} \times (12 + 23) \times 5 \]
\[ A = \frac{1}{2} \times 35 \times 5 \]
\[ A = \frac{175}{2} = 87.5 \text{ square inches} \]
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Calculate the area of the circle: The area \( A \) of a circle is given by the formula:
\[ A = \pi r^2 \]
Given that the diameter of the circle is 5 inches, the radius \( r \) is:
\[ r = \frac{5}{2} = 2.5 \text{ inches} \]
Now, substituting for area:
\[ A = \pi (2.5)^2 \]
\[ A = \pi \times 6.25 = 6.25\pi \text{ square inches} \]
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Calculate the area not inside the circle: The area outside the circle (area of the trapezoid minus area of the circle) is:
\[ A_{not , in , circle} = A_{trapezoid} - A_{circle} \]
\[ A_{not , in , circle} = 87.5 - 6.25\pi \]
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Calculate the probability: The probability \( P \) that a random point placed in the trapezoid is not inside the circle is calculated as:
\[ P = \frac{A_{not , in , circle}}{A_{trapezoid}} = \frac{87.5 - 6.25\pi}{87.5} \]
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Final result: Plugging in the approximate value of \( \pi \approx 3.14 \):
\[ 6.25\pi \approx 6.25 \times 3.14 \approx 19.625 \]
Therefore,
\[ A_{not , in , circle} \approx 87.5 - 19.625 = 67.875 \]
Finally, calculating the probability:
\[ P \approx \frac{67.875}{87.5} \approx 0.774 \] or about \( 77.4% \).
Thus, the probability that a random point placed in the right trapezoid is not inside the circle is approximately \( \frac{67.875}{87.5} \) or \( \approx 0.774 \).