Question

a frame shop charges according to both the amount of framing needed to surround the picture and the amount of glass needed to cover the picture. a. find the area and perimeter of the trapezoid - shaped framed picture to the right. b. identify whether the frame has to do with perimeter or area and the same with the glass. a. the area of the picture is and the perimeter is

Explanation:

Step1: Recall trapezoid area formula

The area formula for a trapezoid is $A=\frac{(a + b)h}{2}$, where $a$ and $b$ are the lengths of the parallel - sides and $h$ is the height. Here, $a = 27$ in, $b = 87$ in, and $h = 16$ in.
\[A=\frac{(27 + 87)\times16}{2}\]

Step2: Calculate the sum of parallel - sides

First, calculate $27+87 = 114$ in. Then, $A=\frac{114\times16}{2}$.
\[A=\frac{1824}{2}=912\text{ in}^2\]

Step3: Recall trapezoid perimeter formula

The perimeter $P$ of a trapezoid is the sum of all its side - lengths. Here, the side - lengths are $27$ in, $34$ in, $87$ in, and $34$ in.
\[P=27 + 34+87 + 34\]

Step4: Calculate the perimeter

\[P=(27 + 87)+(34 + 34)=114 + 68=182\text{ in}\]

Step5: Analyze relationship between frame, glass and picture

The amount of framing needed is related to the perimeter of the trapezoid - shaped picture. The amount of glass needed is related to the area of the trapezoid - shaped picture. The frame is around the outside (perimeter) and the glass covers the flat surface (area). So the frame has to do with the perimeter.

Answer:

a. The area of the picture is $912$ in$^2$ and the perimeter is $182$ in.
b. The frame has to do with perimeter.