Question 1190621: The table below gives the area and perimeter of a rectangle whose length is twice
that of its width (the measurements are in meters).
Width 0 1 2 3 4 5 6
Area of rectangle 0 2 8 18 32 50 72 Perimeter of rectangle 0 6 12 16 24 30 36
a)What are the units in each row of the table?
Row 1 units: ____ Row 2 units: _______ Row 3 units: ________
b)From the table, decide if either area or perimeter could be a linear function of side length. Explain your decision.
c)Write the perimeter as a function of x, where x is the width of the rectangle.
d)Write the area as a function of x, where x is the width of the rectangle.
e)If you found a linear relationship, give its corresponding rate of change and interpret its significance.
f)From the data make two graphs, one showing area as a function of side length, the other showing perimeter as a function of side length. On each graph connect the points.
Answer by CPhill(1987)   (Show Source):
You can put this solution on YOUR website! Here's the breakdown of this rectangle problem:
**a) Units:**
* Row 1 (Width): meters (m)
* Row 2 (Area): square meters (m²)
* Row 3 (Perimeter): meters (m)
**b) Linear Function?**
Perimeter *could* be a linear function of side length (width). As the width increases by 1 meter, the perimeter increases by a consistent 6 meters. Area, however, is *not* linear. The increase in area is not constant as the width increases.
**c) Perimeter as a function of x (width):**
Since the length is twice the width, let:
* x = width
* 2x = length
Perimeter = 2(length + width)
Perimeter = 2(2x + x)
Perimeter = 2(3x)
P(x) = 6x
**d) Area as a function of x (width):**
Area = length * width
Area = (2x) * x
A(x) = 2x²
**e) Linear Relationship:**
The perimeter is a linear function of the width. The rate of change (slope) is 6. This means that for every 1-meter increase in the width of the rectangle, the perimeter increases by 6 meters.
**f) Graphs:**
I can't physically draw graphs here, but I'll describe how they should look:
* **Area vs. Width:** The graph of area vs. width should be a curve (specifically, a parabola opening upwards). The points you would plot are (0,0), (1,2), (2,8), (3,18), (4,32), (5,50), and (6,72).
* **Perimeter vs. Width:** The graph of perimeter vs. width should be a straight line. The points you would plot are (0,0), (1,6), (2,12), (3,18), (4,24), (5,30), and (6,36).
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