Melissa is planning a rectangular vegetable garden with a square patch for tomatoes. She wants the length of the garden to exceed three times the length of the tomato patch by two feet. She also wants the garden’s width to exceed the width of the tomato patch by five feet.Part AMelissa wants to know how the width and length of the garden relate to the length of the square tomato patch. If each side of the tomato patch is x feet, write the functions WG(x) and LG(x) to represent the garden’s width and length, respectively.Part BWrite the function AG(x) representing the area of the garden in terms of x.Part CIf the sides of the square tomato patch are seven feet, find the area of the garden.Part DMelissa decides to reserve a patch in her vegetable garden for growing bell peppers. She wants its width to be half the width of the tomato patch, and its length must exceed the length of the tomato patch by two feet. Write the functions WB(x) and LB(x) representing the width and length, respectively, of the bell pepper patch.Part EWrite the function AB(x) representing the area of the bell pepper patch in terms of x.Part FWrite the function ATB(x) representing the combined area of the tomato patch and the bell pepper patch.Part GYou’ve written functions to represent the area of the tomato patch and the area of the bell pepper patch. Now write the function AR(x) for the remaining planting area in the garden.Part HIf Melissa wants the area of the bell pepper patch to be 31.5 square feet, find the area of the remaining space in the garden after planting tomatoes and bell peppers.

[tex]\begin{gathered} AG(x)=WG(x)\cdot LG(x) \\ \rightarrow AG(x)=(x+5)(3x+2) \\ \rightarrow AG(x)=3x^2+2x+15x+10 \\ \\ \Rightarrow AG(x)=3x^2+17x+10 \end{gathered}[/tex]

Part C

Let's evaluate x = 7 in AG(x)

[tex]\begin{gathered} AG(7)=3(7^2)+17(7)+10 \\ \rightarrow AG(7)=276 \end{gathered}[/tex]

Thereby, the area of the garden would be 276 square feet

Part D

The functions would be:

[tex]\begin{gathered} WB(x)=\frac{x}{2} \\ LB(x)=x+2 \end{gathered}[/tex]

Part E

[tex]\begin{gathered} AB(x)=WB(x)\cdot LB(x) \\ \rightarrow AB(x)=(\frac{x}{2})(x+2) \\ \\ \Rightarrow AB(x)=\frac{x^2}{2}+x \end{gathered}[/tex]

Part F

[tex]\begin{gathered} ATB(x)=x^2+AB(x) \\ \rightarrow ATB(x)=x^2+\frac{x^2}{2}+x \\ \\ \Rightarrow ATB(x)=\frac{3}{2}x^2+x \end{gathered}[/tex]

Part G

[tex]\begin{gathered} AR(x)=AG(x)-ATB(x) \\ \rightarrow AR(x)=3x^2+17x+10-\frac{3}{2}x^2-x \\ \\ \Rightarrow AR(x)=\frac{3}{2}x^2+16x+10 \end{gathered}[/tex]

Part H

We have a function for the area of the bell pepper patch in terms of x, the measurement of the lenght and width of the tomato patch. This is:

[tex]AB(x)=\frac{x^2}{2}+x[/tex]

We know the value of this area. This way, we can solve the equation for x,

[tex]31.5=\frac{x^2}{2}+x\rightarrow63=x^2+2x\rightarrow x^2+2x-63=0[/tex]

Using the cuadratic formula, and ignoring non-positive results, we'll get that

[tex]x=7[/tex]

Now, plugging in this value in AR(x),

[tex]\begin{gathered} AR(7)=\frac{3}{2}(7^2)+16(7)+10 \\ \Rightarrow AR=195.5 \end{gathered}[/tex]

This way, we can conclude that the remaining space in the garden after planting tomatoes and bell peppers is 195.5 square feet