Question

How we can apply Quadratic Functions in this situation ?

Given a rectangular water tank with Volume = Length x Breadth x Height
= `x*2*(k-x)`

Given k is a constant and the maximum volume of the water tank is `50m^3` . Find the value of `k` .

Answer 1

`k = 10" m"`

Explanation:

Given: `V = (x" m")(2" m")(k" m"-x" m")`

Multiply the factors, put them in standard form, and mark as equation [1]:

`V = -2x^2" m"^3+2kx" m"^3" [1]"`

This is standard form

`V = ax^2+bx + c`

where `a = -2" m", b = 2k" m"^2, and c = 0`

Because "a" is negative, we know that the maximum will occur at the vertex. The x coordinate of the vertex is:

`x = -b/(2(a))`

Substitute the values for "a" and "b":

`x = -(2k" m"^2)/(2(-2" m")`

`x = k/2" m"`

We know that `V = 50" m"^3` at the vertex:

Substitute the known values into equation [1]:

`50" m"^3= -(2" m")(k/2" m")^2+(2k" m"^2)(k/2" m")`

Multiply terms:

`50" m"^3= -2k^2/4" m"^3+k^2" m"^3`

Combine like terms:

`50" m"^3= k^2/2" m"^3`

`100" m"^2=k^2" m"^2`

`k = 10" m"`

Substitute into equation [1]:

`V = -2x^2+20x" m"^3" [2]"`

Here is a graph of of equation [2]:

graph{-2x^2+20x [-14.01, 22.02, 39.21, 57.23]}

Please observe that 50 is the maximum.