Chapter 2.4, Problem 22E

To determine

Write an equation of the parabola .Find the dimensions and corresponding area of one possible rectangle .What dimensions result in the maximum area.

Answer to Problem 22E

Equation of the parabola is $y=-x^2+7x.$

Maximum area is 12.25 and dimension is 3.5.

Explanation of Solution

Given information: The area of a rectangle is modeled by the graph where y is the.

area (in square meters) and x is the width (in meters).

  

Calculation:

Given a point and x-intercept p and q ,then the equation is , $$

  $y=a(x-p)(x-q).$........(1)

From the graph $p=0\text{and}q=7$ and point is (1,6) then,

  $\begin{array}{l}y=a(x-0)(x-7).\\ 6=a(1)(1-7).\\ 6=-6a.\\ a=-1.\end{array}$

Putting the value of $a=-1,p=0,q=7$ in equation (1),then the equation of the para bola is,

  $\begin{array}{l}y=-(x-0)(x-7).\\ y=-x^2+7x.\end{array}$

Identify the coefficients.

  $a=-1,b=7,c=0.$

Since $a<0$ ,parabola opens down and function has a maximum value.

To find maximum value, calculate coordinates of the vertex.

  $\begin{array}{l}x=-b/2a=-7/2\times (-1)=-7/(-2)=3.5.\\ $ y=-{(3.5)}^2+7\times (3.5)=12.75.$\end{array}$

So ,

Equation of the parabola is $y=-x^2+7x.$

Maximum area is 12.25 and dimension is 3.5