Question

7.)
a =
8.)
a =
challenge problems: find the area of the shaded regions
9.)
a = sq. units
10.)
a = sq. units

Explanation:

7. Area of a circle

The formula for the area of a circle is \(A = \pi r^{2}\). Given the diameter \(d = 14\) in, the radius \(r=\frac{d}{2}=\frac{14}{2}=7\) in.
Substitute \(r = 7\) into the formula:
\(A=\pi\times(7)^{2}=\pi\times49 = 153.86\) in² (using \(\pi = 3.14\))

8. Area of a triangle

The formula for the area of a triangle is \(A=\frac{1}{2}bh\). Here, the base \(b = 10\) mm and the height \(h = 5\) mm.
Substitute \(b = 10\) and \(h = 5\) into the formula:
\(A=\frac{1}{2}\times10\times5=25\) mm²

9. Area of a composite - figure

The composite figure is made up of two rectangles.
For the first rectangle (top - left): One side is \(3\) and the other side is \(7 - 4=3\), so its area \(A_{1}=3\times3 = 9\)
For the second rectangle (bottom - right): \(A_{2}=7\times4=28\)
The total area \(A=A_{1}+A_{2}=9 + 28=37\) sq.units

10. Area of a shaded region (circle - square)

The formula for the area of a circle is \(A_{circle}=\pi r^{2}\), with \(r = 10\), so \(A_{circle}=\pi\times(10)^{2}=314\) (using \(\pi = 3.14\))
The formula for the area of a square is \(A_{square}=s^{2}\), with \(s = 2\), so \(A_{square}=2\times2=4\)
The area of the shaded region \(A=A_{circle}-A_{square}=314 - 4=310\) sq.units

Answer:

1. \(153.86\) in²
2. \(25\) mm²
3. \(37\)
4. \(310\)