Mathematics, 21.07.2019 19:00 cjjjjjjjjjjjjj

Answer all 5 parts. 1.melissa 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 2 feet. write the functions wa(x) and la(x) representing the width and length, respectively, of the bell pepper patch. type response. 2.write the function aa(x) representing the area of the bell pepper patch in terms of x. 3.write the function aar(x) representing the combined area of the tomato patch and the bell pepper patch. 4.you’ve written functions to represent the area of the tomato patch and the area of the bell pepper patch. now write the function aa(x) for the remaining planting area in the garden. 5.if 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.

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Answer from: mszeew

A function

from a set A to a set B is defined as a relation that assings to each element

in the set A exactly one element

in the set B. The set A is called the domain of the function while the set B is the range. So we have five statements and need to find some functions. Melissa decides to reserve a patch in her vegetable garden for growing bell peppers. If each side of the tomato patch is

feet, then we have a square patch as shown in the Figure below.

1.a) Write the function Wa(x) representing the width of the bell pepper patch.

We know that she wants its width to be half the width of the tomato patch. Let

be the width of the tomato patch, then the function that matches this statement is:

$\boxed{Wa(x)=\frac{x}{2}}$

1.b) Write the function La(x) representing the length of the bell pepper patch.

In this case Melissa wants its length to exceed the length of the tomato patch by 2 feet. To do this we enlarge the length of the tomato patch 2 feet. Therefore the function is the following:

$\boxed{La(x)=x+2}$

2. Area of the bell pepper patch in terms of x.

Given that the bell pepper patch is a rectangle, then the area of a rectangle is the product of the length and width. So:

$A=(\frac{x}{2})(x+2) \\ \\ \therefore \boxed{A=\frac{x(x+2)}{2}}$

3. Combined area of the tomato patch and the bell pepper patch.

This function is the sum of both the area of the tomato patch and the bell pepper patch. So:

$Aar(x)=x^{2}+\frac{x(x+2)}{2} \rightarrow Aar(x)=x^{2}+ \frac{x^{2}}{2}+x \rightarrow Aar(x)=\frac{3x^{2}}{2}+x \\ \\ \therefore \boxed{Aar(x)=\frac{x(3x+2)}{2}}$

4. Write the function Aa(x) for the remaining planting area in the garden.

The remaining planting area in the garden are the rectangles in red. So we need to subtract the width of the bell pepper patch from the width of the tomato patch and multiply it by 2. In mathematical language this is given by:

$Aa(x)=2(x-\frac{x}{2}) \rightarrow Aa(x)=x$

5. Find the area of the remaining space in the garden after planting tomatoes and bell peppers.

Given that Melissa wants the area of the bell pepper patch to be 31.5 square feet, then it is true that:

$31.5=\frac{x(x+2)}{2} \rightarrow x^{2}+2x-64=0 \\ \\ solving \ for \ x: \\ x=7.06$

Therefore the area of the remaining space is:

$\boxed{Aa(7.06)=7.06ft^{2}}$

Answer all 5 parts. 1.melissa decides to reserve a patch in her vegetable garden for growing bell p