Question

NO LINKS!! URGENT HELP PLEASE!!!

1. Make a table showing the length, width, and area for every rectangle with a perimeter of 20 meters and whole-number side lengths. Describe some patterns that you observe in the table.


2. Make a graph of the data(length, area). Describe the shape of the graph.


3. How does the pattern in the table appear in the graph?

Answer 1

1. See below.

2. See attachment.

3. See below.

Explanation:

Question 1

Definition of variables:

• Let w be the width of the rectangle.
• Let l be the length of the rectangle.

A rectangle has two pairs of parallel sides of equal length.

Therefore, the equation for the perimeter of a rectangle is:

`P=2(w+l)`

If the perimeter is 20 meters then:

`\implies 2(w+l)=20`

`\implies w+l=10`

If the side lengths are whole numbers, then the length and width are restricted to whole numbers from 1 to 9 (inclusive).

The equation for the area of a rectangle is:

`A =w\cdot l`

Table

`\begin{array}c\cline{1-3}\vphantom{\frac12}\textsf{width, m}&\textsf{length, m}&\textsf{Area, m$^2$}\\\cline{1-3}\vphantom{\frac12}1&9&9\\\cline{1-3}\vphantom{\frac12}2&8&16\\\cline{1-3}\vphantom{\frac12}3&7&21\\\cline{1-3}\vphantom{\frac12}4&6&24\\\cline{1-3}\vphantom{\frac12}5&5&25\\\cline{1-3}\vphantom{\frac12}6&4&24\\\cline{1-3}\vphantom{\frac12}7&3&21\\\cline{1-3}\vphantom{\frac12}8&2&16\\\cline{1-3}\vphantom{\frac12}9&1&9\\\cline{1-3}\end{array}`

• As the width increases by 1 m, the length decreases by 1 m.
• The values for area are symmetric about the point when width and length are the same.
• The step pattern of the values of area in the table are 7, 5, 3, 1, ...
  This is the step pattern of a quadratic function that opens downwards.

Question 2

To create a graph of the data, plot the length along the x-axis and the area along the y-axis. Plot the following points:

• (9, 9), (8, 16), (7, 21), (6, 24), (5, 25), (4, 24), (3, 21), (2, 16) and (1, 9).

The shape of the graph is a parabola that opens downwards.

Its vertex is (5, 25) and its axis of symmetry is x = 5.

Question 3

This step pattern of 7, 5, 3, 1, ... is apparent in the graph; as you move away from the vertex in the x-direction, the graph goes down by 7, then 5, then 3, etc. This is the step pattern of a parabola that opens downwards.

There is an axis of symmetry on the graph at the side length value when the width and length are the same (x = 5).