David has 200m of fencing to enclose three identicle rectangular pens as shown in the figure. Let x be the length of each pen.
a) express the total area of the pens in terms of x
My answer: 2x^2-100x
(It is correct according to the answer key)
b) find the maximum total area of the pens
????????
Thankyou!!
the maximum area is at the vertex of the parabola.
That is always at the axis of symmetry, x = -b/2a
In this case, that is at x = 100/4 = 25
2*25^2 - 100*25 = -1250
???
Your function cannot be correct, since it is a parabola opening upward, and thus has no maximum. The coefficient of x^2 must be negative for the function to have a maximum.
To find the maximum total area of the pens, you can use calculus. We know that the total area of the pens is given by the expression 2x^2 - 100x.
To find the maximum, we need to find the critical points of this quadratic function. The critical points occur where the derivative is equal to zero or is undefined.
First, let's find the derivative of the function. Differentiating 2x^2 - 100x with respect to x, we get:
dA/dx = 4x - 100
Setting this derivative equal to zero and solving for x:
4x - 100 = 0
4x = 100
x = 25
So, x = 25 is a critical point.
Next, we can check the endpoints of the possible domain. Since we are fencing three identical rectangular pens, the length of each pen cannot be negative. Therefore, the domain of the function is x ≥ 0.
Checking the endpoint, x = 0, we have:
A(0) = 2(0)^2 - 100(0) = 0
So, the critical point x = 25 and the endpoint x = 0 are the only two candidates for the maximum area.
To determine which one gives the maximum area, we need to compare the values of the total area at these points.
A(x = 0) = 0
A(x = 25) = 2(25)^2 - 100(25) = 1250 - 2500 = -1250
From these calculations, we can see that there is no maximum area. The total area of the pens is always negative for this given scenario, which would not make practical sense.
Therefore, it is likely that there is a mistake either in the problem statement or in our assumption that the three identical rectangular pens should be enclosed by the given fencing.