how many pounds of rock will you need to fill an area 25 ft by 35 ft to a depth of 2 in. If the rock weighs 1050 lb cubic yard?
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Answer:
Step-by-step explanation:
This is a pretty basic related rates problem. I'm going to go through this just like I do in class when I'm teaching it to my students.
We see we have a snowball, which is a sphere. We are talking about the surface area of this sphere which has a formula of
In the problem we are given diameter, not radius. What we know about the relationship between a radius and a diameter is that
d = 2r so
Now we can have the equation in terms of diameter instead of radius. Rewriting:
which simplifies to
and a bit more to
(the 4's cancel out by division). Now that is a simple equation for which we have to find the derivative with respect to time.
Now let's look at the problem and see what we are given as far as information.
The rate at which the surface area changes is -3.8, and we are looking for , the rate at which the diameter is changing, when the diameter is 13. Filling in:
and solving for the rate at which the diameter is changing:
and divide to get
Obviously, the negative means that the diameter is decreasing.