Expert: Ahmed Salami - 10/30/2004

Question
Please help me with these 5 problems that I am stuck on. You don't need to do all 5, do what you can do.

Thanks!

1.Suppose we want to construct a box with a top.We want the box to have a capacity of 300 cubic units. For economic reasons, we would like to use the least amount of material to construct the box. Let the dimensions of the base be x by y and the height be h. Draw the box. Write the equation of the surface area of the box as a function of the two variables x and y.

2.A Cylindrical tank 4 ft in diameter fills with water at the rate of 10 ft/s. Express the depth of the water in the tank as a function of the time t in seconds.Assume the tank is empty at time t=0.

3. A store owner bought x dozen toy dolls at a cost of $4.00 per dozen, and sold them at $85 a piece.Express the profit P(in dollars) as a function of x.

4.Two Sides of a rectangle like along the axes as shown at the right. one vertex is a point on the line 3x+4y=24.
a.Express the area A of the rectangle as a function of the x-coordinate of P.
b.What is the domain of the area function?
c.What is the maximum area of A?

5.Triangle OAB is an isosceles triangle with vertex O at the origin and vertices A and B on the part of the parabola y=8-x^2 that is above the x-axis.
a.Express the area of the triangle as a function of the x-coordinate of A.
b.What is the domain of the area function?
c.Find the maximum area.

Answer
Hi Tin,
Sorry about the delay. I'll , however, try my best.
1)The volume of the box would be
V = xyh = 300
Therefore,
h = 300/xy
But the surface area of the box would be
S = 2xy + 2xh + 2yh
S = 2xy + 2x(300/xy) + 2y(300/xy)
S = 2xy + 600/y + 600/x

2)Let the height of the cylinder be H and the height of the level of water at any time t be h. If the depth of water at any time t is z, then
z = H - h
But dh/dt = 10
Integrating gives h = 10t considering that h = 0 when
t = 0. With no further information available,
z = H - 10t

3)If x dozen toys are bought at $4 per dozen, total cost price = $4x
In total there are 12x toys. At $85 each, they would be sold for a total of 12x(85) = $1020x
Therefore, the profit P as a function of x would be
P = 1020x - 4x
 = $1016x

4)a)The area A = xy where (x,y)is the coordinate of the point on the line.
From 3x + 4y = 24
y = (24 - 3x)/4
y = 6 - 3x/4
A = x(6 - 3x/4)
 = 6x - 3x^2/4
b)For the domain, x can take all real values.
c)The maximum area occurs when dA/dx = 0
dA/dx = 6 - 3x/2
equating to zero gives x = 4
The maximum area is thus
A = 6(4)- 3(16)/4
 = 24 - 12 = 12

5)a)The area A = 1/2(2x)y = xy where (x,y)and (-x,y) are the coordinates of the points on the parabola.
y = 8 - x^2
A = x(8 - x^2)
 = 8x - x^3
b)For the domain, x can take all real values.
c)The maximum area occurs when dA/dx = 0
dA/dx = 8 - 3x^2
equating to zero gives x = sqrt(8/3)
The maximum area is thus
A = 8sqrt(8/3) - [sqrt(8/3)]^3
 = 8.7

I hope you understand them all. You can always get back to me on anything unclear.
Regards.