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]
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
