Question

2 - 59. cairo wants to create a graph that represents the heights and bases of all rectangles that have an area of 36 square units. he started by drawing the rectangles a, b, c, and d at right. examine the dimensions (length and width) of each rectangle. homework help a. copy the graph at right onto graph paper. then match the letter of each rectangle above with a point on the graph. which point is not matched? b. what are the base, height, and area for the unmatched point? c. why should the unmatched point not be on cairos graph? d. find the dimensions of three more rectangles that have areas of 36 square units. at least one of your examples should have dimensions that are not integers. place a new point on the graph for each new rectangle you find. e. connect all of the points representing an area of 36 square units. describe the resulting graph.

Explanation:

Step1: Recall area formula for rectangle

The area of a rectangle is $A = base\times height$. Given $A = 36$. So we need to find pairs of numbers (base and height) such that their product is 36.

Step2: Analyze rectangles A - D

For rectangle A: Count the number of unit - squares along the base and height. Suppose base $b$ and height $h$. If we assume the grid represents unit - squares, we find the dimensions. Similarly for B, C and D.

Step3: Find the unmatched point

We check which point on the graph does not correspond to the dimensions of rectangles A - D.

Step4: Determine base, height and area for unmatched point

If the unmatched point has coordinates $(x,y)$, then the base is $x$ and height is $y$ and area $A=x\times y$.

Step5: Find non - integer dimension rectangles

We look for pairs of non - integer numbers that multiply to 36. For example, if $base = 4.5$ and $height = 8$, then $4.5\times8 = 36$.

Step6: Connect points for new rectangles

We plot the points representing the new rectangles with non - integer dimensions on the graph and connect them.

Answer:

a. Copy the graph: (Instructions to copy the graph are not provided in detail here as it requires manual drawing. But one should redraw the given graph on graph - paper). Then match the letters: Analyze the dimensions of rectangles A, B, C and D by counting the unit - squares. For example, if rectangle A has a base of 6 units and a height of 6 units (since $6\times6 = 36$).
b. For the unmatched point, assume the point has coordinates $(x,y)$. The base is $x$, height is $y$ and area is $x\times y$. If we assume the points on the graph represent (height, base) pairs, and we find that a point like $(9,4)$ might be matched with a rectangle in the set A - D. If there is a point not corresponding to the known rectangles, say a point $(3.6,10)$ (non - integer example), base = 3.6, height = 10 and area $=3.6\times10 = 36$.
c. The unmatched point should not be on Cairo's graph if it does not represent the dimensions of the rectangles A - D. It could be because its dimensions do not match the ones of the drawn rectangles.
d. Examples of non - integer dimension rectangles: $(4.5,8)$ since $4.5\times8 = 36$, $(7.2,5)$ since $7.2\times5 = 36$. Place new points on the graph for these rectangles.
e. After placing all the points for rectangles with area 36 (including non - integer dimension ones), connect the points. The resulting graph will show all rectangles with an area of 36 square units, and may form a pattern or a set of scattered points depending on the nature of the dimensions.