Question

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: Assume base $b_1$ and height $h_1$. We need to check if $b_1\times h_1=36$. Similarly for B, C and D.

Step3: Find dimensions for new rectangles

The factor - pairs of 36 are $(1,36),(2,18),(3,12),(4,9),(6,6),(9,4),(12,3),(18,2),(36,1)$. We can use these to create new rectangles.

Step4: Answer part a

Copy the graph. Then, for each rectangle, we match the letter with a point on the graph by checking the base - height combinations. The point not matched will be determined by elimination.

Step5: Answer part b

The base, height and area relationships are based on $A = base\times height$. For the unmatched point, we can find the base and height such that their product is 36.

Step6: Answer part c

The unmatched point should not be on the graph if it does not represent a valid base - height combination for a rectangle with area 36 among the ones already plotted or considered.

Step7: Answer part d

Using the factor - pairs of 36, we can find three more rectangles. For example, if base $b = 1$ and height $h = 36$; base $b=2$ and height $h = 18$; base $b = 3$ and height $h=12$.

Step8: Answer part e

Connect the points representing rectangles with area 36. The resulting graph will show the relationship between the bases and heights of rectangles with a fixed area of 36.

Answer:

a. After copying the graph and matching rectangles A - D with points on the graph, identify the non - matched point by checking base - height combinations.
b. For the unmatched point, if the area formula $A=base\times height = 36$ is used, we can find the base and height values.
c. The unmatched point should not be on the graph if its base - height combination does not match any of the valid rectangles with area 36 among those considered.
d. Three more rectangles with area 36 can have dimensions $(1,36),(2,18),(3,12)$ (or other non - used factor pairs of 36).
e. The resulting graph will show a set of points representing the bases and heights of rectangles with area 36, and connecting them may show a pattern (e.g., a hyperbola - like shape in a non - formal sense for the relationship $xy = 36$ where $x$ is the base and $y$ is the height).