To find the area of the yellow shaded region, we need to subtract the area of the purple circle from the area of the large triangle.
Step 1: Find the area of the triangle.
The formula for the area of a triangle is:
$$ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} $$
From the diagram:
* The base of the triangle is $14$ inches.
* The height of the triangle is $10$ inches.
Calculate the area:
$$ \text{Area}_{\text{triangle}} = \frac{1}{2} \times 14 \times 10 $$
$$ \text{Area}_{\text{triangle}} = 7 \times 10 $$
$$ \text{Area}_{\text{triangle}} = 70 \text{ square inches} $$
Step 2: Find the area of the circle.
The formula for the area of a circle is:
$$ \text{Area} = \pi \times r^2 $$
From the diagram:
* The radius ($r$) of the circle is $3$ inches.
Calculate the area (using $\pi \approx 3.14$):
$$ \text{Area}_{\text{circle}} = \pi \times 3^2 $$
$$ \text{Area}_{\text{circle}} = \pi \times 9 $$
$$ \text{Area}_{\text{circle}} \approx 3.14 \times 9 $$
$$ \text{Area}_{\text{circle}} \approx 28.26 \text{ square inches} $$
*(Note: If using the exact value of $\pi$, the area is $9\pi$.)*
Step 3: Subtract the circle's area from the triangle's area.
$$ \text{Shaded Area} = \text{Area}_{\text{triangle}} - \text{Area}_{\text{circle}} $$
$$ \text{Shaded Area} = 70 - 28.26 $$
$$ \text{Shaded Area} = 41.74 \text{ square inches} $$
If leaving the answer in terms of $\pi$:
$$ \text{Shaded Area} = 70 - 9\pi \text{ square inches} $$
Final Answer:
The area of the shaded region is approximately
41.74 square inches (or exactly $70 - 9\pi$ sq in).
Parent Tip: Review the logic above to help your child master the concept of geometric probability problems.
