College Algebra Word Problems?

Question

I have two questions that I can't figure out and if you would explain how to do them that would be great!

1. A person wants to make an enclosed play pen using 60 meters of fence. One of the sides is against the house (thus, not requiring any fencing). What are the dimensions of the play pen to get the maximum area?

2. A boat can go 12 mph in calm water. If the boat goes down a river 45 miles and back up the river 45 miles it takes him 8 hours. What is the current of the river?

Thanks in advance!

Answer 1

1. The equation for the fence is going to be x+x+(60-x). That would make the area x(60-x).

X=the length of one side of fence
x=the length of the opposite side of the fence
60-x=the length of the side opposite the house

so, plug in some numbers and see what you get.

2. 45/(12 +x) + 45/(12-x) = 8

x=the current

Answer 2

Only one problem per question please.

2. A boat can go 12 mph in calm water. If the boat goes down a river 45 miles and back up the river 45 miles it takes him 8 hours. What is the current of the river?

Let
c = speed of current in river
r = average rate round trip
t = time round trip
d = distance one way

r = 2d/t = 2*45/8 = 90/8 = 45/4

r = 2d/t = 2d/[d/(12 + c) + d/(12 - c)] = 2/[1/(12 + c) + 1/(12 - c)]
45/4 = 2/[1/(12 + c) + 1/(12 - c)]
45[1/(12 + c) + 1/(12 - c)] = 2*4
45[(12 - c) + (12 + c)] = 8(12 + c)(12 - c)
45*24 = 8(144 - c²)
135 = 144 - c²
c² = 9
c = 3

The speed of the current in the river is 3 mph.

Answer 3

1. I'm going to assume that this play pen has four sides, one of which is taken up by the house, so actually three sides.

Lets call one side x, and its opposite side also has to be x in order for this to be rectangular. The third side therefore must be (60 - 2x).

Area = length * width = (x)(60-2x)
And we want to maximize. What I would do is graph this and then find a point that is the highest within your domain. The x value at which you get this vertex is the width. You can solve for length by just plugging in x into (60-2x)

If you knew a little bit of calculus, you could take the derivative of this function and set it equal to zero to find your x value.