Question

1. the length of a soccer field is 31 m shorter than two times its width.

a) draw a diagram and label the length and width with algebraic expressions. define your variables with let statements.
b) determine a simplified expression representing the perimeter of the soccer field.
c) determine a simplified expression for the area of the soccer field.
d) what is the new perimeter if you double the width?
e) what is the new area if you double the width?
f) what are the perimeter and area if the width is 68m?

Explanation:

Step1: Define variables

Let \( w\) be the width of the soccer field. Then the length \( l = 2w-31\).

Step2: Calculate perimeter formula

The perimeter formula of a rectangle is \( P=2(l + w)\). Substitute \( l = 2w - 31\) into the formula:

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Step3: Calculate area formula

The area formula of a rectangle is \( A=l\times w\). Substitute \( l = 2w - 31\) into the formula:

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Step4: New width and new perimeter

If the width is doubled (\( w_{new}=2w\)), then \( l_{new}=2(2w)-31 = 4w-31\).
The new perimeter \( P_{new}=2((4w - 31)+2w)\)

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Step5: New area

The new area \( A_{new}=(4w - 31)\times2w\)

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Step6: Substitute \( w = 68\)

For perimeter: \( P=6\times68-62=408 - 62=346\)
For area: \( A=2\times68^{2}-31\times68=2\times4624-2108=9248-2108 = 7140\)

Answer:

a) Let \( w\) (width), \( l = 2w-31\) (length). Diagram: rectangle with one side labeled \( w\) and adjacent side labeled \( 2w - 31\).
b) \( 6w-62\)
c) \( 2w^{2}-31w\)
d) \( 12w - 62\)
e) \( 8w^{2}-62w\)
f) Perimeter \( 346m\), Area \( 7140m^{2}\)