Calculus Applied Solutions

Plus other problems with solutions by: Jerry C. DeKeyser
Golden Mean Problem
     
Derive the "Golden Mean" by comparing two similar rectangles that share a common side. (Not a calculus problem.)
001 		Maximum Box Volume Problem
         Drawing of sheet of metal ready to be folded into box
Use a 10”x10” piece of metal to construct a box without top by removing the corners and folding the sides up. What are the dimensions for a box with the maximum possible volume?
193 		Maximum Prism Volume Problem
         Prism with triangular end and length of 10 inches.
The ends of a prism are isosceles triangles with two 4 inch sides. The sides of the prism are rectangles with a length of 10 inches. What is the maximum possible volume of this prism?
177
Moving Ladder Problem
         Drawing of ladder with x and y coordinates
The bottom of a 13 foot ladder is moving away from the wall at 3 feet per second. How fast is the top of the ladder moving down when the bottom of the ladder is 5 feet from the wall?
152
Maximum Area Problem
      

A track with semicircular ends and a rectangular center has a perimeter of 440 yards. What are the dimensions of the rectangular section if that section has maximum possible area?
196
Distance Between Problem
         Drawing of map with two ships and their coordinates
A ship is 20 miles north of a sailboat. The ship is sailing south at 40 mph and the sailboat is sailing east at 20 mph. What will be the closest distance between the ship and sailboat?
197
Water Filling Cone Problem
         Drawing of water filling a cone with r and h coordinates
A cone has a height of 6 inches and a radius of 2 inches. Water is poured in at rate of 23 cubic inch per second. How fast does the water level rise when the water level is 4 inches deep?
Use volume formula: V = ⅓ π r2 h
153
Decreasing Shadow Problem
         Drawing of overlapping triangles with x and y coordinates
Bill is 6 foot tall and he walks towards a street lamp that is 30 foot high. What is the rate Bill's shadow is decreasing if the rate Bill walks towards the light is 4 feet per second?

154
Changing Angle Problem
         Drawing baseball diamond with dimensions
A baseball player is running 24 feet per second to third base and an umpire is standing on home plate looking at the runner. When the runner is 30 feet from third base, how fast is the angle between third base and the umpire's line of sight to the runner changing?
157
Boat Moving Rate Problem
     Drawing of man pulling rope attached to boat and x, y and z coordinated.
One end of a rope is attached to a boat. The other end is pulled over a pulley that is 5 feet high. How fast is the boat moving to shore when the rope is pulled 2 feet per second and the length of the rope from the boat to the pulley is 13 feet?
156A
Rope Pulling Rate Problem
     Drawing of man pulling rope attached to boat and x, y and z coordinated.
One end of a rope is attached to a boat. The other end is pulled over a pulley that is 5 feet high. How fast is the rope being pulled when the boat is being pulled 2 feet per second and the length of the rope from the boat to the pulley is 13 feet?
156B
Distance of Boat Problem
     Drawing of man pulling rope attached to boat and x, y and z coordinated.
One end of a rope is attached to a boat. The other end is pulled over a pulley that is 8 feet high. How far is the boat from the shore when the rope is being pulled 2 feet per second and the boat is moving to shore at 2415 feet per second?
156C
Lighthouse Beacon Problem
     Drawing lighthoue by shore with x, y and z coordinates.
The beacon on a lighthouse revolves at a rate of 10pi radians per a minute. The lighthouse is one mile north of a straight east to west shore. What speed does its light sweep across the shore two miles from the lighthouse?
156
Cars at Intersection Problem
      Drawing a parked car and a moving car at an intersection witn x and y coordinates.
One car is parked 3 miles from an intersection. Another car is on a perpendicular street moving toward the intersection at 30 mph. How fast is the distance between the two cars decreasing when the moving car is 4 miles from the intersection?
158R
Spherical Balloon Problem
        Drawing of a balloon with r radius indicated.
Air is being pumped into a spherical balloon at a constant rate of 36π cubic inches per minute. How fast is the radius of the balloon increasing when the radius is 3 inches?
Volume of sphere: V = 43π r³
159R
Exponential Growth Problem
If an algae doubles every two days and it grows exponentially, what will its size be after seven days, if is it starts with a size of 1,000,000? (Not a calculus problem.)
Use formula: f(t) = f(0) ekt -- k is a constant;t is time in days.
332
Carbon-14 Decay Problem
If Carbon-14 (C14) has a half life of 5730 years what be the percentage of C14 will be present 1000 years after the death of an organism? (Not a calculus problem.)
Use formula: f(t) = f(0) ekt -- k is a constant; t is time in years.
333
Carbon-14 Dating Problem
If Carbon-14 (C14) has a half life of 5730 years and a bone contains only 1/10th of the original C14, then how long ago did the organism die? (Not a calculus problem.)
Use formula: f(t) = f(0) ekt -- k is a constant; t is time in years.
334