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Building Rectangles
A set of rectangles can be formed out of $1 \mathrm{~cm}$ squares on centimetre grid paper. Martina draws the first rectangle with dimensions $1 \mathrm{~cm} \times 2 \mathrm{~cm}$, the second $2 \mathrm{~cm} \times 4 \mathrm{~cm}$, and the third $3 \mathrm{~cm} \times 6 \mathrm{~cm}$.
? If this pattern continues, what function can be used to model the relationship between width and area?
A. Use centimetre grid paper to draw the next four rectangles in the pattern. Use your diagrams to extend and complete the table.
B. Calculate the first differences for the Perimeter-and Area columns. know?
c. Is the relation between width and area linear or nonlinear? How do you
know? The reldtion thip befween width and area i
D. Determine the second differences for the Area column. What do they tell you about the relationship between width and area?
E. Create a scatter plot of area versus width. Draw a curve of good fit. Does the shape of your graph support your answer to part D? Explain.
F. What is the relationship between length and width for each rectangle?
G. Write the function that models the relationship between width and area.
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Answer from Sia

Posted 3 months ago

Solution by Steps

step 1

Observe the pattern in the dimensions of the rectangles

step 2

The width of the rectangles is increasing by 1 cm each time, and the length is twice the width

step 3

The area of a rectangle is given by the formula $A = \text{length} \times \text{width}$

step 4

Since the length is twice the width, we can express the area as a function of width: $A(w) = 2w \times w = 2w^2$

step 5

To draw the next four rectangles, increase the width by 1 cm each time and double it to find the length

step 6

Calculate the perimeter and area for each rectangle and fill in the table

step 7

Calculate the first differences for the perimeter and area columns by subtracting the previous value from the current value

step 8

Determine if the relationship between width and area is linear by looking at the first differences. If they are constant, the relationship is linear

step 9

Calculate the second differences for the area column by subtracting the previous first difference from the current first difference

step 10

If the second differences are constant, the relationship is quadratic

step 11

Create a scatter plot of area versus width and draw a curve of best fit

step 12

Analyze the shape of the graph to support the findings from the second differences

step 13

The relationship between length and width for each rectangle is that the length is always twice the width

step 14

Write the function that models the relationship between width and area as $A(w) = 2w^2$

Answer

The function that models the relationship between width and area is $A(w) = 2w^2$.

Key Concept

Quadratic Relationship

Explanation

The relationship between width and area is quadratic because the area is proportional to the square of the width, as shown by the function $A(w) = 2w^2$.

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