The volume of a glass can be modelled by rotating the function [math]f\left(x\right)=\frac{x^4}{25}[/math] ([i]x[/i], [i]f(x)[/i] in cm) around the [i]y[/i]-axis. The volume of the glass should be 500 ml. Estimate the height of the glass.[br][br][b]Hint:[/b] Use the formula [math]\pi\cdot\int_a^b\left(f^{^{-1}}\left(x\right)\right)^2dx[/math] to calculate the volume of revolution around the [i]x[/i]-axis of the portion of [i]f(x)[/i] in the [i]x[/i]-interval [i][a, b][/i].

[table][tr][td]1.[/td][td]Enter the function[math]f\left(x\right)=\frac{1}{25}\cdot x^4[/math] into the [i]Input Bar[/i] and press [i]Enter[/i]. [/td][/tr][tr][td][br][/td][td][b]Note:[/b] The graph of [i]f(x)[/i] will be displayed in the [i]Graphics View.[/i][/td][/tr][tr][td]2.[/td][td]Calculate the inverse of [i]f(x)[/i] using the command [math]Invert(f)[/math].[/td][/tr][tr][td]3.[/td][td]Press the [i]More[/i] button and select [i]Add label[/i] to label the inverse of [i]f(x)[/i] with [i]g[/i].[/td][/tr][tr][td]4.[/td][td]Use the formula from the hint to calculate the volume of revolution around the [i]x[/i]-axis of [i]f(x)[/i] in the [i]x[/i]-interval [i][0,h][/i], where [i]h[/i] is the height of the glass. [/td][/tr][tr][td][/td][td]Enter the command [math]a=\pi\cdot Integral\left(g^2,0,h\right)[/math] into the [i]Input Bar [/i]and press [i]Enter[/i].[/td][/tr][tr][td]5.[/td][td]As the volume should be 500 ml, solve the equation [math]\pi\cdot Integral\left(g^2,0,h\right)=500[/math].[/td][/tr][tr][td][/td][td]Therefore use the label you gave to the Integral and enter the command [math]Solutions\left(a=500\right)[/math].[/td][/tr][tr][td]6.[/td][td]Press the [img]https://wiki.geogebra.org/uploads/thumb/6/66/Numeric_toggle_button.png/24px-Numeric_toggle_button.png[/img] numeric toggle button to show the numeric solution. As result, the height of the glass is about 13.16 cm.[/td][/tr][tr][td][/td][td][b]Note:[/b] It is also possible to use more commands in one input. For example Steps 4 and 5 can be done by entering [math]Solutions\left(Integral\left(g^2,0,h\right)=500\right)[/math]. [i]GeoGebra CAS Calculator[/i] will automatically add a right parenthesis [i])[/i] when entering your command with a left parenthesis [i]([/i].[/td][/tr][/table]