Apunch glass is in the shape of a hemisphere with a radius of 5 cm. if the punch is being poured into the glass so that the change in height of the punch is 1,5 cm/sec, at what rate is the exposed area of the punch changing when the height of the punch is 2 cm.

28.27 cm/s

Step-by-step explanation:

Though Process:

Solution:

the area of the exposed circle is

the rate of change of this area can be, (using chain rule)

we can call this Eq(2)

what we are really concerned about is how changes as the punch is being poured into the bowl i.e

So we need another formula: Using the property of hemispheres and pythagoras theorem, we can use:

and rearrage the formula so that a is the subject:

now we can derivate a with respect to h to get

simplify

we can put this in Eq(1) in place of

and since we know

and now we use substitute this . in Eq(2)

simplify,

This is the rate of change of area, this is being asked in the quesiton!

Finally, we can put our known values:

from the question

what