Chapter 3.4, Problem 9AYU

To determine

To calculate:

The largest area

Answer to Problem 9AYU

The largest area is $2,000,000meter{s}^2$ .

Explanation of Solution

Given information:

  

Calculation:

Know that maximum point of parabola is $x=-b/2a$ .

First to find the largest area, you need to find a function that graphs the area with respect to $x.$ Since you ae given the length and width, the area function is found by multiplying them together.

  $\begin{array}{l}area=x.\left(4000-2x\right)\\ area=4000x-2x^2\end{array}$

So, now the maximum point on the parabola since it signifies the largest area,

  $\begin{array}{l}\text{Max}\text{Po}\mathrm{int}:\left(-\frac{b}{2a},f\left(-\frac{b}{2a}\right)\right)\\ \Rightarrow \left(-\frac{4000}{-4},f\left(-\frac{4000}{-4}\right)\right)\\ \Rightarrow \left(1000,2000000\right)\end{array}$

You can also find this point by putting the equation in $h-k$ form by completing the square: $y=a.{\left(x-h\right)}^2+k$

  $\begin{array}{l}y=-2x^2+4000x\\ y=-2.\left(x^2-2000x\right)\\ y=-2.\left(x^2-2000x+{\left(\frac{-2000}{2}\right)}^2\right)-\left(-2.{\left(\frac{-2000}{2}\right)}^2\right)\\ \therefore y=-2.{\left(x-1000\right)}^2+2000000\\ \text{point:}\left(1000,2000000\right)\end{array}$

Hence, the maximum area is $10,000y{d}^2$ .