Questions: A rectangular field is to be enclosed by 840 feet of fence. One side of the field is a building, so fencing is not required on that side. If (x) denotes the length of one side of the rectangle perpendicular to the building, determine the function in the variable (x) giving the area (in square feet) of the fenced-in region. Area, as a function of (x = x(840-2x)) Determine the domain of the area function. Enter your answer using interval notation. Domain of area function =

A rectangular field is to be enclosed by 840 feet of fence. One side of the field is a building, so fencing is not required on that side. If (x) denotes the length of one side of the rectangle perpendicular to the building, determine the function in the variable (x) giving the area (in square feet) of the fenced-in region. Area, as a function of (x = x(840-2x)) Determine the domain of the area function. Enter your answer using interval notation. Domain of area function =
Transcript text: A rectangular field is to be enclosed by 840 feet of fence. One side of the field is a building, so fencing is not required on that side. If $x$ denotes the length of one side of the rectangle perpendicular to the building, determine the function in the variable $x$ giving the area (in square feet) of the fenced-in region. Area, as a function of $x=$ $\square$ $x(840-2 x)$ Determine the domain of the area function. Enter your answer using interval notation. Domain of area function $=$ $\square$ Submit Answer 7. [1/1 Points] DETAILS MY NOTES
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Solution

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Solution Steps

Step 1: Understanding the Problem

We need to determine the domain of the area function for a rectangular field that is enclosed by 840 feet of fence. One side of the field is a building, so fencing is not required on that side. The area function is given as \( A(x) = x(840 - 2x) \).

Step 2: Setting Up the Equation

The total length of the fence used is 840 feet. Since one side is a building, the fencing is required for the other three sides. Let \( x \) be the length of the side perpendicular to the building. The remaining fencing will be used for the two sides parallel to the building, each of length \( x \), and the side opposite the building, which is \( 840 - 2x \).

Step 3: Determining the Domain

To find the domain of the area function \( A(x) = x(840 - 2x) \), we need to ensure that the lengths of the sides are positive. Therefore, \( x \) must be greater than 0 and less than \( 420 \) (since \( 2x \) must be less than 840).

Final Answer

The domain of the area function is \( (0, 420) \).

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