Calculus Test

Front 1

A sheet of cardboard 3 ft. by 4 ft. will be made into a box by cutting equal-sized squares from each corner and folding up the four edges. What will be the dimensions of the box with largest volume ?

Back 1

V=3.03 ft cubed

Front 2

Find two nonnegative numbers whose sum is 9 and so that the product of one number and the square of the other number is a maximum.

Back 2

108

Front 3

Find the dimensions of the rectangle of largest area which can be inscribed in the closed region bounded by the x-axis, y-axis, and graph of y=8-x3

Back 3

7.60

Front 4

If V is the volume of the cube with edge length x and the cube expands as time passes. Find dV/dt in terms of dx/dt. If the length of the edge is increasing at a constant speed 1 cm/s, how fast is the volume changing when the edge length is 20 cm?

Back 4

1200 cm cubed / sec

Front 5

At 7:00 A.M. a truck is 60 miles due north of a car. The truck is traveling south at a constant speed of 40 mph, while the car is traveling east at 60 mph. How fast is the distance between the car and the truck changing at 7:30 A.M.?

Back 5

4mph

Front 6

A person is sitting on a bench in a park and watching a balloon rising up in the air 100 m away from him. Balloon is rising up at a constant speed of 5m/sec. A person moves his head in order to keep balloon in sight. How fast does the person move his head when the balloon is at the height of 50 m?

Back 6

1/25 radians

Front 7

Sand is being dumped into a conical pole whose height is 1/2 the radius of its base. Suppose sand is being pumped at a rate 5 cubic meters per minute.
A) How fast is the height of the pile increasing when it is 9 meters height?
B) How fast is the area of the base increasing at this moment?
C) How fast is the circumference of the base increasing at this moment?
D) Will the height be increasing more slowly, more rapidly or at steady pace as time goes on?

Back 7

a) 5/324 pi
b) 10/9
c) 5/81
d) 5/4pi h squared
bigger h, smaller dh/dt, increasing more slowly