To solve the problem, let's carefully break it down step by step.
1. Initial Area of the Field:
The farmer’s initial rectangular field measures 100 feet by 150 feet. Therefore, the initial area [tex]\( A_{\text{initial}} \)[/tex] is:
[tex]\[
A_{\text{initial}} = 100 \text{ ft} \times 150 \text{ ft} = 15000 \text{ ft}^2
\][/tex]
2. Increased Area:
The farmer wants to increase the area by [tex]\( 20\% \)[/tex]. Therefore, the new area [tex]\( A_{\text{new}} \)[/tex] is:
[tex]\[
A_{\text{new}} = A_{\text{initial}} \times 1.20 = 15000 \text{ ft}^2 \times 1.20 = 18000 \text{ ft}^2
\][/tex]
3. New Dimensions:
The farmer will increase both the length and the width by the same amount [tex]\( x \)[/tex]. Therefore, the new length will be [tex]\( 100 + x \)[/tex] and the new width will be [tex]\( 150 + x \)[/tex].
4. Equation for the New Area:
The equation representing the new area in terms of [tex]\( x \)[/tex] will be:
[tex]\[
(\text{new length}) \times (\text{new width}) = A_{\text{new}}
\][/tex]
Substituting the new dimensions and the new area:
[tex]\[
(100 + x)(150 + x) = 18000
\][/tex]
This equation needs to be among the answer choices. Let's match it with the given options:
a. [tex]\((100+2x)(150+2x)=18000\)[/tex] - This is incorrect because it has [tex]\(2x\)[/tex] added to each dimension instead of [tex]\(x\)[/tex].
b. [tex]\(2(100+2x)+2(150+x)=15000\)[/tex] - This is incorrect because it represents a perimeter-like formula, not an area.
c. [tex]\((100+x)(150+x)=18000\)[/tex] - This represents our derived equation correctly.
d. [tex]\((100+x)(150+x)=15000\)[/tex] - This is incorrect because the right-hand side should be 18000, as per the need to increase the area by [tex]\(20\%\)[/tex].
e. [tex]\((x+100)(x+150)=(1.2) 15000\)[/tex] - Simplifying the right-hand side:
[tex]\[
1.2 \times 15000 = 18000
\][/tex]
Therefore, this equation also correctly gives us the increased area.
From the options, the correct equations are:
- [tex]\( (100 + x)(150 + x) = 18000 \)[/tex] (option c)
- [tex]\( (x + 100)(x + 150) = (1.2) 15000 \)[/tex], which simplifies to [tex]\( (x + 100)(x + 150) = 18000 \)[/tex] (option e).
Thus, the correct selections are:
- c. [tex]\((100+x)(150+x)=18000\)[/tex]
- e. [tex]\((x+100)(x+150)=(1.2) 15000\)[/tex]
