Answer:
![\begin{array}{cc}Length \ of \ Tomato \ Patch \ (in \ feet)&Area \ of \ Vegetable \ Garden \ (in \ square \ feet)\\\ [6.25]&338\\6.5&[242.625]\\ \ [6.75]&147.25\\7&[51.875]\end{array}](https://tex.z-dn.net/?f=%5Cbegin%7Barray%7D%7Bcc%7DLength%20%5C%20of%20%5C%20Tomato%20%5C%20Patch%20%5C%20%28in%20%5C%20feet%29%26Area%20%5C%20of%20%5C%20Vegetable%20%5C%20Garden%20%5C%20%28in%20%20%5C%20square%20%5C%20feet%29%5C%5C%5C%20%5B6.25%5D%26338%5C%5C6.5%26%5B242.625%5D%5C%5C%20%20%5C%20%5B6.75%5D%26147.25%5C%5C7%26%5B51.875%5D%5Cend%7Barray%7D)
Step-by-step explanation:
The date from the given table, is expressed as follows;
Area of Vegetable Garden (in square feet); 338, ___, 147.25, ___
Length of Tomato Patch (in feet); ___, 6.5, ___, 7
The length of the tomato patch increases with decrease in the area of the garden
Given that the length of the tomato patch is the independent variable, we can have;
Length of Tomato Patch (in feet); 6.25, 6.5, 6.75, 7
Therefore, for an increase in the length of the tomato patch from 6.25 to 6.75, (a change of Δl = 0.5) the area of the vegetable garden decreased from 338 to 147.25 which is a decrease of ΔA = 190.75
Therefore, the width of the tomato patch, w = ΔA/Δl = 190.75/0.5 = 381.5
The width of the tomato patch, w = 381.5 ft.
The relationship between the total area, TA, the area of the vegetable garden, <em>A</em>, and the length of the tomato patch, <em>l</em>, is therefore, given as follows;
A = TA - 381.5·l
338 = TA - 381.5×6.25
TA = 338 + 381.5×6.25 = 2,722.375
Therefore, when l = 6.5, we get
A = 2,722.375 - 381.5×6.5 = 242.625
When l = 7, we get
A = 2,722.375 - 381.5×7 = 51.875
Therefore, we get;
![\begin{array}{cc}Length \ of \ Tomato \ Patch \ (in \ feet)&Area \ of \ Vegetable \ Garden \ (in \ square \ feet)\\\ [6.25]&338\\6.5&[242.625]\\ \ [6.75]&147.25\\7&[51.875]\end{array}](https://tex.z-dn.net/?f=%5Cbegin%7Barray%7D%7Bcc%7DLength%20%5C%20of%20%5C%20Tomato%20%5C%20Patch%20%5C%20%28in%20%5C%20feet%29%26Area%20%5C%20of%20%5C%20Vegetable%20%5C%20Garden%20%5C%20%28in%20%20%5C%20square%20%5C%20feet%29%5C%5C%5C%20%5B6.25%5D%26338%5C%5C6.5%26%5B242.625%5D%5C%5C%20%20%5C%20%5B6.75%5D%26147.25%5C%5C7%26%5B51.875%5D%5Cend%7Barray%7D)