Question

Agriculture A farmer grows tomatoes and cucumbers in the field shown. The annual cost

of growing tomatoes is \(.27 per square foot. The annual cost of growing cucumbers is \).10 per square foot. Let x represent the width (in feet) of the tomato portion of the field.

1. In terms of x, what is the area of the tomato portion? of the cucumber portion?
2. Write and simplify an expression in terms of x for the annual cost of growing both crops.
3. Find the annual cost of growing both crops if the width of the tomato portion is 350 feet
4. 
   

Short answer

(a) Area of tomato portion is$500$square ft and

Area of cucumber portion is $500(800-x)$square ft.

(b) The simplified expression is $\$(85x+40000)$

(c) The annual cost is $\$\text{69750}$

Step-by-step solution

(a)Step 1   Apply the concept of terms and like terms

In an algebraic expression, terms are separated by plus or minus signs.

The term coefficient refers to the number in front of the variable in any term of the expression.

A constantterm is the number which have no variable associated with it.

Like terms are the terms which have identical variable parts with same exponent on it. Constants are considered to be like terms as well.

Distributive property: $a\left(b+c\right)=a·b+a·c$

  Given information and formula used

The tomato and cucumber portions have dimensions as shown in figure,

Formula:

Area$=$Length $\times$Breadth

  Calculation

The length of tomato is $500$ft

And its breath is $x$ft.

So, the area of the tomato is $500x$ft.

The length of the cucumber is $500$ft.

And its breath is $=(800-x)$ft.

So, the area of the cucumber is $=500(800-x)$square ft.

(b)Step 1   Apply the concept of terms and like terms

In an algebraic expression, terms are separated by plus or minus signs.

The term coefficient refers to the number in front of the variable in any term of the expression.

A constantterm is the number which have no variable associated with it.

Like terms are the terms which have identical variable parts with same exponent on it. Constants are considered to be like terms as well.

Distributive property: $a\left(b+c\right)=a·b+a·c$

  Given information and formula used

Annual cost of growing tomatoes and cucumber is $\$·27$and $\$·10$

As found in part (a), the areas of tomatoes and cucumber portions are $500x$square feet and $500(800-x)$square feet respectively.

  Calculation

So, the annual cost of growing both crops is

$\begin{array}{l}\$(0\cdot 27)\left(500x\right)+\$(0\cdot 10)\left(500\right(800-x\left)\right)\\ =\$135x+\$50(800-x)\\ =\$135x+\$(40000-50x)\\ =\$(135x+40000-50x)\\ =\$(85x+40000)\end{array}$

(c)Step 1   Apply the concept of terms and like terms

In an algebraic expression, terms are separated by plus or minus signs.

The term coefficient refers to the number in front of the variable in any term of the expression.

A constantterm is the number which have no variable associated with it.

Like terms are the terms which have identical variable parts with same exponent on it. Constants are considered to be like terms as well.

Distributive property: $a\left(b+c\right)=a·b+a·c$

  Given information and formula used

Width of tomato portion $x=350$ft .

As found in part (b),

The annual cost is given by $\$(85x+40000)$

  Calculation

Thus for $x=350$ft

The annual cost $=\$(85\times 350+40000)$

$\begin{array}{l}=\$(29750+40000)\\ =\$69750\end{array}$