Chapter 2.1, Problem 59E

To determine

To find:Twonumbers whose sum is given and product is maximum.

Answer to Problem 59E

The product of two numbers whose sum is 110, is maximum when they are 55 and 55

Explanation of Solution

Given data:

The sum of two numbers is 110.

Concept used:

The vertex of downwards opening quadratic function is maxima of the function.

Calculations:

Let x and z be two numbers whose sum is 110 and product is maximum. We denote the product of numbers by y. Thus,

  $x+z=110$  .........(1)

  $y=xz$  .........(2)

From results (1) and (2), we have

  $y=x\left(110-x\right)=-x^2+110x$  .........(3)

As $y=-x^2+110x$ is a downwards opening parabola therefore, its maximum occurs at the vertex $\left(x\text{,}{y}_{\text{maximum}}\right)$ .

Now we obtain standard form of quadratic function $y=-x^2+110x$ and find its vertex.

  $y=-x^2+110x=-\left(x^2-2\times 55x+{55}^2\right)+{55}^2$

  $y=-{\left(x-55\right)}^2+3025$  .........(4)

We know the vertex of quadratic function (parabola) $y=-a{\left(x-\alpha \right)}^{\text{2}}+\beta$ is $\left(\alpha ,\beta \right)$ that is for $x=\alpha$ , $y_{\text{maximum}}=\beta$ . Therefore, the vertex of quadratic function (4) is $\left(55,3025\right)$ that implies $y_{\text{maximum}}=3025$ for $x=55$ .

That shows, the product $y=xz$ of two numbers x, zis maximum when $x=55$ and other number $z=110-55=55$ .

Thus, the product of two numbers whose sum is 110, is maximum when they are 55 and 55.

Conclusion:

The product of two numbers whose sum is 110, is maximum when they are 55 and 55.