Respuesta :

carloggrlj

Answer:

[tex]h'=\frac{dh}{dr}=-\frac{2}{r^3\pi}[/tex]

Step-by-step explanation:

Assuming the dough is of cylindrical shape and that the volume must stay the same the equation for the volume of the cylinder is the following:

[tex]V=r^2\pi h[/tex]

where V is the volume, r the radius and h the height of the cylinder. If you get h to the left hand side you get the following equation:

[tex]h=\frac{V}{r^2\pi}[/tex]

To find the rate of change of the height you need to derive the above equation with respect to r:

[tex]h'=-\frac{2}{r^3\pi}[/tex]

MrRoyal

Rate of change is simply how much a quantity changes, over another.

The expression for the rate of change of the dough in terms of height h and radius r is [tex]\mathbf{h' = \frac{-2}{\pi r^3}}[/tex]

From the complete question, we have:

[tex]\mathbf{V = \pi r^2h}[/tex]

Next, we make h the subject

[tex]\mathbf{h = \frac{V}{\pi r^2}}[/tex]

Rewrite as:

[tex]\mathbf{h = \frac{V}{\pi}r^{-2}}[/tex]

Differentiate with respect to r

[tex]\mathbf{h' = -2\times \frac{V}{\pi}r^{-2-1}}[/tex]

[tex]\mathbf{h' = -2\times \frac{V}{\pi}r^{-3}}[/tex]

Rewrite as:

[tex]\mathbf{h' = \frac{-2V}{\pi r^3}}[/tex]

Remove V, to leave the answer in terms of r and h

[tex]\mathbf{h' = \frac{-2}{\pi r^3}}[/tex]

Hence, the expression for the rate of change of the dough in terms of height h and radius r is [tex]\mathbf{h' = \frac{-2}{\pi r^3}}[/tex]

Read more about rates of change at:

https://brainly.com/question/12786410