A farmer has 100 m of fencing to enclose a rectangular pen.
Which quadratic equation gives the area ($A$) of the pen, given its width $(w)$?
A. $A(w)=w^2-50 w$
B. $A(w)=w^2-100 w$
C. $A(w)=50 w-w^2$
D. $A(w)=100 w-w^2
Answer (2)
Express the perimeter in terms of width w and length l : 2 w + 2 l = 100 .
Solve for the length l in terms of the width w : l = 50 − w .
Express the area A as a function of width w : A ( w ) = w ( 50 − w ) .
Simplify the area equation: A ( w ) = 50 w − w 2 . The final answer is A ( w ) = 50 w − w 2 .
Explanation
Problem Analysis Let's analyze the problem. We have a rectangular pen with a perimeter of 100 m. We need to find a quadratic equation that expresses the area A of the pen in terms of its width w .
Finding the Length in Terms of Width Let the width of the rectangular pen be w and the length be l . The perimeter is given by 2 w + 2 l = 100 . We can solve for l in terms of w : 2 l = 100 − 2 w l = 50 − w
Calculating the Area The area of the rectangular pen is given by A = l w . Substituting the expression for l in terms of w into the area equation, we get: A = ( 50 − w ) w A = 50 w − w 2
Expressing Area as a Function of Width Therefore, the area as a function of w is A ( w ) = 50 w − w 2 .
Final Answer The quadratic equation that gives the area A of the pen, given its width w , is A ( w ) = 50 w − w 2 .
Examples
Imagine you're designing a garden and have a limited amount of fencing. This problem helps you determine the dimensions that will give you the largest possible area for your plants. By understanding how the width affects the area, you can optimize your garden layout to maximize the growing space. This principle applies to various real-world scenarios, such as optimizing the layout of solar panels to capture the most sunlight or designing efficient storage spaces.
The quadratic equation that expresses the area A of the rectangular pen in terms of its width w is A ( w ) = 50 w − w 2 . Therefore, the correct answer is option C. This equation shows how the area depends on the width of the pen given a fixed perimeter.
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