AP Calculus Studyguide for Test #6 (3.7,2.6)Optimization & Related Rates

Show all work on another sheet. You may use calculators. Give exact answers if possible or round to 3 decimal places. (You may solve graphically but confirm on your paper analytically.)

Formulas: Volume of a cone =1/3πr2hVolume of a Sphere =4/3 πr3

Volume of a Cylinder = πr2hSurface Area of a Sphere = 4πr

Optimization (Maximizing and Minimizing)

  1. A farmer plans to fence a rectangular pasture adjacent to a river. The pasture must contain 1/2 square miles in order to provide enough grass for the herd. What dimensions would require the least amount of fencing if no fencing is needed along the river?
  2. What is the smallest perimeter possible for a rectangle whose area is 100cm2? What are its dimensions?
  3. A manufacturer wants to design an open-top box made by cutting congruent squares of side length x from the corners of a 10in by 15 in sheet of tin and bending up the sides. How large should the squares be to make the box hold as much as possible?
  4. Find 2 positive integers whose sum is 12 and whose product is as large as possible.
  5. Find the point on f(x)=√x that is closest to (6,0).

Related Rates

  1. A 16 ft ladder is leaning against a vertical wall. If the bottom of the ladder is pulled along the ground away from the wall at a constant rate of 2 ft/sec, how fast will the top of the ladder be moved down the wall when the ladder is 6ft from the wall?
  2. A tank filled with water is in the shape of an inverted cone 50 ft high with a circular base on top whose radius is 20 feet. Water is running out of the bottom of the tank at a constant rate of 7 ft3/min. How fast is the water level falling when the water is 9 feet deep? (Hint: You have to find a relationship between r and h in the original figure and substitute.)
  3. A spherical balloon is to be deflated so that the volume decreases at a rate of 5 ft3/min. How fast is the diameter of the balloon decreasing when the radius is 3 feet?
  4. A point moves along the curve y=2x2-1 in such a way that the y value is decreasing at a rate of 4 units per second. At what rate is x changing when x = 2.5?
  5. A pebble is dropped into a calm pond, causing ripples in the form of concentric circles. The radius of the outer ripple is increasing at a constant rate of 5 feet per second. When the radius is 7 feet, at what rate is the total area of the disturbed water changing?

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AP Calculus Studyguide for Test #6 (3.7,2.6)Optimization & Related Rates

Show all work on another sheet. You may use calculators. Give exact answers if possible or round to 3 decimal places. (You may solve graphically but confirm on your paper analytically.)

Formulas: Volume of a cone =1/3πr2hVolume of a Sphere =4/3 πr3

Volume of a Cylinder = πr2hSurface Area of a Sphere = 4πr

Optimization (Maximizing and Minimizing)

  1. A farmer plans to fence a rectangular pasture adjacent to a river. The pasture must contain 1/2 square miles in order to provide enough grass for the herd. What dimensions would require the least amount of fencing if no fencing is needed along the river?
  2. What is the smallest perimeter possible for a rectangle whose area is 100cm2? What are its dimensions?
  3. A manufacturer wants to design an open-top box made by cutting congruent squares of side length x from the corners of a 10in by 15 in sheet of tin and bending up the sides. How large should the squares be to make the box hold as much as possible?
  4. Find 2 positive integers whose sum is 12 and whose product is as large as possible.
  5. Find the point on f(x)=√x that is closest to (6,0).

Related Rates

  1. A 16 ft ladder is leaning against a vertical wall. If the bottom of the ladder is pulled along the ground away from the wall at a constant rate of 2 ft/sec, how fast will the top of the ladder be moved down the wall when the ladder is 6ft from the wall?
  2. A tank filled with water is in the shape of an inverted cone 50 ft high with a circular base on top whose radius is 20 feet. Water is running out of the bottom of the tank at a constant rate of 7 ft3/min. How fast is the water level falling when the water is 9 feet deep? (Hint: You have to find a relationship between r and h in the original figure and substitute.)
  3. A spherical balloon is to be deflated so that the volume decreases at a rate of 5 ft3/min. How fast is the diameter of the balloon decreasing when the radius is 3 feet?
  4. A point moves along the curve y=2x2-1 in such a way that the y value is decreasing at a rate of 4 units per second. At what rate is x changing when x = 2.5?
  5. A pebble is dropped into a calm pond, causing ripples in the form of concentric circles. The radius of the outer ripple is increasing at a constant rate of 5 feet per second. When the radius is 7 feet, at what rate is the total area of the disturbed water changing?