Question

1. think! diagram with 54 in, 30 in, 49 in measurements

Explanation:

To determine the area of the given composite figure, we can break it down into a trapezoid and a quarter - circle (since the curved part seems to be a quarter - circle with radius \(r = 30\) in).

Step 1: Area of the trapezoid

The formula for the area of a trapezoid is \(A_{trapezoid}=\frac{(a + b)h}{2}\), where \(a\) and \(b\) are the lengths of the two parallel sides and \(h\) is the height.
Here, \(a = 30\) in, \(b = 49\) in, and \(h = 54\) in.
\[A_{trapezoid}=\frac{(30 + 49)\times54}{2}=\frac{79\times54}{2}=79\times27 = 2133\space in^{2}\]

Step 2: Area of the quarter - circle

The formula for the area of a circle is \(A_{circle}=\pi r^{2}\). For a quarter - circle, the area is \(A_{quarter - circle}=\frac{1}{4}\pi r^{2}\)
Given \(r = 30\) in and taking \(\pi\approx3.14\)
\[A_{quarter - circle}=\frac{1}{4}\times3.14\times30^{2}=\frac{1}{4}\times3.14\times900 = 3.14\times225=706.5\space in^{2}\]

Step 3: Total area of the composite figure

To find the total area, we add the area of the trapezoid and the area of the quarter - circle.
\[A = A_{trapezoid}+A_{quarter - circle}=2133 + 706.5=2839.5\space in^{2}\]

If we assume that the problem is to find the area of the figure (since it's a common problem with such a diagram), the final answer is \(\boldsymbol{2839.5}\space square\space inches\) (or depending on the value of \(\pi\) used, if we use \(\pi=\frac{22}{7}\))

Recalculating with \(\pi=\frac{22}{7}\) for the quarter - circle:

\[A_{quarter - circle}=\frac{1}{4}\times\frac{22}{7}\times30^{2}=\frac{1}{4}\times\frac{22}{7}\times900=\frac{22\times225}{7}=\frac{4950}{7}\approx707.14\space in^{2}\]
\[A = 2133+\frac{4950}{7}=\frac{2133\times7 + 4950}{7}=\frac{14931+4950}{7}=\frac{19881}{7}\approx2840.14\space in^{2}\]

Answer:

To determine the area of the given composite figure, we can break it down into a trapezoid and a quarter - circle (since the curved part seems to be a quarter - circle with radius \(r = 30\) in).

Step 1: Area of the trapezoid

The formula for the area of a trapezoid is \(A_{trapezoid}=\frac{(a + b)h}{2}\), where \(a\) and \(b\) are the lengths of the two parallel sides and \(h\) is the height.
Here, \(a = 30\) in, \(b = 49\) in, and \(h = 54\) in.
\[A_{trapezoid}=\frac{(30 + 49)\times54}{2}=\frac{79\times54}{2}=79\times27 = 2133\space in^{2}\]

Step 2: Area of the quarter - circle

The formula for the area of a circle is \(A_{circle}=\pi r^{2}\). For a quarter - circle, the area is \(A_{quarter - circle}=\frac{1}{4}\pi r^{2}\)
Given \(r = 30\) in and taking \(\pi\approx3.14\)
\[A_{quarter - circle}=\frac{1}{4}\times3.14\times30^{2}=\frac{1}{4}\times3.14\times900 = 3.14\times225=706.5\space in^{2}\]

Step 3: Total area of the composite figure

To find the total area, we add the area of the trapezoid and the area of the quarter - circle.
\[A = A_{trapezoid}+A_{quarter - circle}=2133 + 706.5=2839.5\space in^{2}\]

If we assume that the problem is to find the area of the figure (since it's a common problem with such a diagram), the final answer is \(\boldsymbol{2839.5}\space square\space inches\) (or depending on the value of \(\pi\) used, if we use \(\pi=\frac{22}{7}\))

Recalculating with \(\pi=\frac{22}{7}\) for the quarter - circle:

\[A_{quarter - circle}=\frac{1}{4}\times\frac{22}{7}\times30^{2}=\frac{1}{4}\times\frac{22}{7}\times900=\frac{22\times225}{7}=\frac{4950}{7}\approx707.14\space in^{2}\]
\[A = 2133+\frac{4950}{7}=\frac{2133\times7 + 4950}{7}=\frac{14931+4950}{7}=\frac{19881}{7}\approx2840.14\space in^{2}\]