Question

1. find the area of the shaded region:
2. if a rectangles length is 7 more than 4 times the width and the perimeter is 194 mm, what are the dimensions of the rectangle?

Explanation:

Step1: Find the area of the rectangle

The area formula of a rectangle is \(A = l\times w\). Given \(l = 14\) in and \(w=9\) in.
\(A_{rectangle}=14\times9 = 126\) in²

Step2: Find the area of the circle

The diameter of the circle \(d = 9\) in, so the radius \(r=\frac{d}{2}=\frac{9}{2}=4.5\) in. The area formula of a circle is \(A=\pi r^{2}\). Taking \(\pi = 3.14\)
\(A_{circle}=3.14\times(4.5)^{2}=3.14\times20.25 = 63.585\) in²

Step3: Find the area of the shaded region

The area of the shaded region \(A = A_{rectangle}-A_{circle}\)
\(A=126 - 63.585=62.415\) in²

Step1: Set up variables

Let the width of the rectangle be \(w\) mm. Then the length \(l=(4w + 7)\) mm.

Step2: Use the perimeter formula

The perimeter formula of a rectangle is \(P = 2(l + w)\). Given \(P = 194\) mm.
Substitute \(l=(4w + 7)\) into the formula: \(194=2((4w + 7)+w)\)
First, simplify the equation inside the parentheses: \(194=2(5w + 7)\)
Then, distribute the 2: \(194 = 10w+14\)

Step3: Solve for \(w\)

Subtract 14 from both sides: \(194-14=10w\), so \(180 = 10w\)
Divide both sides by 10: \(w = 18\) mm

Step4: Solve for \(l\)

Substitute \(w = 18\) into \(l=(4w + 7)\)
\(l=4\times18+7=72 + 7=79\) mm

Answer:

The area of the shaded region is \(62.415\) in²