Question

a rancher has 100 meters of fencing to enclose part of a field in the shape of a rectangle. the equation below represents ( a ), the total area in square - meters, where ( x ) is the length in meters of the rectangle. ( a=x(50 - x) ). the rancher wants to find the length and width that would maximize the area enclosed by the fence. he is completing the square as shown below, but he still needs to find the value of ( c ). ( a=-x^{2}+50x ), ( a=-(x - 25)^{2}+c ). find the correct value of ( c ) to complete the ranchers work. drag and drop the correct numbers into the boxes to complete the sentences below. the value of ( c ) is . a maximum area of square meters is obtained when the length is meters and the width is meters. -625 -100 -50 -25 25 50 100 625 2,500

Explanation:

Step1: Expand the square

We start with $A=-(x - 25)^2+c=-(x^{2}-50x + 625)+c=-x^{2}+50x-625 + c$. Comparing with $A=-x^{2}+50x$, we can see that $-625 + c = 0$. Solving for $c$ gives $c = 625$.

Step2: Analyze the quadratic - function

The quadratic function $A=-(x - 25)^2+625$ is in vertex - form $y=a(x - h)^2+k$, where $a=-1\lt0$, $h = 25$, and $k = 625$. Since $a\lt0$, the parabola opens downwards and the vertex $(h,k)$ represents the maximum of the function.

Step3: Find the dimensions

The length $x = 25$ meters. If the length of the rectangle is $x$ and the perimeter equation (not fully shown but from the context of the area formula $A=x(50 - x)$) implies that if $x$ is the length, then the width is $50 - x$. When $x = 25$, the width is also $50-25 = 25$ meters.

Answer:

$625$; $625$; $25$; $25$