Calculus in Physical Chemistry

[jeffinnabi]

Hello All,

I just started my first semester of Physical Chemistry and received a packet with 10 questions and Im stumped on two of them. Any help with these would be appreciated, not looking for answers just how to go about solving the problem.

1. A Home Depot customer wants to enclose a garden inside of a rectangular (possibly square) area with wire mesh fencing material. She has enough money to purchase 100 feet of fencing. If one of the boundaries is the brook that runs through her backyard, how should she configure the fencing so as to enclose the largest possible area?
-I am assuming that she only needs to cover 3 sides of the rectangle with fencing because of the brook on the fourth side. I understand what its asking but not how to solve.

2. From time t=0.0 hours to time t=3.25 hours a person travels at a speed given by the following equation:
s(t)=20.0 miles/hour^2 * t + 1.0 miles/t+10.0 hour
What is the distance traveled?
the speed is a function of the time here but im not sure what i should be doing.

I got the other eight problems over the course of yesterday and today but still cannot get these two. Thank you to anyone who can help.

 

[galactus]

1. A Home Depot customer wants to enclose a garden inside of a rectangular (possibly square) area with wire mesh fencing material. She has enough money to purchase 100 feet of fencing. If one of the boundaries is the brook that runs through her backyard, how should she configure the fencing so as to enclose the largest possible area?
-I am assuming that she only needs to cover 3 sides of the rectangle with fencing because of the brook on the fourth side. I understand what its asking but not how to solve.

Yes, that is correct.

The perimeter is \(\displaystyle 2x+y=100\)...........[1]

The area is \(\displaystyle A=xy\)...........[2]

The area is what we must maximize.

Solve [1]for y and sub into [2]

From [1], \(\displaystyle y=100-2x\)

\(\displaystyle A=x(100-2x)\)

\(\displaystyle A=-2x^{2}+100x\)

Now, differentiate, set to 0 and solve for x. y will follow. Sub the x and y values into A=xy to find the max area.

Or, you can solve it without calculus by using the vertex of a parabola formula: \(\displaystyle x=\frac{-b}{2a}, \;\ y=c-\frac{b^{2}}{4a}\)