Calculus Answers

How do I find the recursive definition for an arithmetic sequence

for each of the following curves, make a sketch of the curve and find:
I. the gradient function
II. the gradient of the tangent at the given point.
a. y=X2 + 3 at point (2,7)
b. y=y2 + k at point (1,1 + k)

1. A page is to contain 24 square inches of print. The margins at top and bottom are 1.5 inches, at the side is 1 inches. Find the most economical dimension of the page.
2. A norman window consists of a rectangle surrounded by a semi circle. What shape gives the most light for a GIVEN perimeter?
3. Find the dimension of the largest rectangular building that can be placed on a right triangular lot, facing one of the perpendicular sides.

23. An open-topped cylindrical pot is to have volume 250 cm3.The material for the bottom of the pot costs 4 cents per cm2; that for its curved side costs 2cents per

cm2. What dimensions will minimize the total cost of this pot?

24. A storage container is to be made in the form of a right circular cylinder and
have a volume of 28p m3. Material for the top of the container costs $5 per
square metre and material for the side and base costs $2 per square metre.
What dimensions will minimize the total cost of the container?

19. A rectangular box has a square base with edge length x of at least 1 unit. The
total surface area of its six sides is 150 square units.
(a) Express the volume V of this box as a function of x.
(b) Find the domain of V (x).
(c) Find the dimensions of the box in part (a) with the greatest possible
volume. What is this greatest possible volume?

20. An open-top box is to have a square base and a volume of 13500 cm3. Find the dimensions of the box that minimize the amount of material used.

21. Find the dimension of the right circular cylinder of maximum volume that can
be inscribed in a right circular cone of radius R and height H.

22. A hollow plastic cylinder with a circular base and open top is to be made and
10 m2 plastic is available. Find the dimensions of the cylinder that give the
maximum volume and find the value of the maximum volume.

16. The top and bottom margins of a poster are each 6 cm, and the side margins
are each 4 cm. If the area of the printed material on the poster (that is,
the area between the margins) is fixed at 384 cm2, find the dimensions of the
poster with the smallest total area.

17. Each rectangular page of a book must contain 30 cm2 of printed text, and each
page must have 2 cm margins at top and bottom, and 1 cm margin at each
side. What is the minimum possible area of such a page?

18. Maya is 2 km offshore in a boat and wishes to reach a coastal village which
is 6 km down a straight shoreline from the point on the shore nearest to the
boat. She can row at 2 km/hr and run at 5 km/hr. Where should she land
her boat to reach the village in the least amount of time?

13. A boy starts walking north at a speed of 1.5 m/s, and a girl starts walkng
west at the same point P at the same time at a speed of 2 m/s. At what rate
is the distance between the boy and the girl increasing 6 seconds later?

14. A police car, approaching right-angled intersection from the north, is chasing
a speeding SUV that has turned the corner and is now moving straight east.
When the police car is 0.6 km north of intersection and the SUV is 0.8 km
east of intersection, the police determine with radar that the distance between
them and the SUV is increasing at 20 km/hr. If the police car is moving at
60 km/hr at the instant of measurement, what is the speed of the SUV?


15. A farmer has 400 feet of fencing with which to build a rectangular pen. He
will use part of an existing straight wall 100 feet long as part of one side of
the perimeter of the pen. What is the maximum area that can be enclosed?

10. The height of a rectangular box is increasing at a rate of 2 meters per second
while the volume is decreasing at a rate of 5 cubic meters per second. If
the base of the box is a square, at what rate is one of the sides of the base
decreasing, at the moment when the base area is 64 square meters and the
height is 8 meters?

11. Sand is pouring out of a tube at 1 cubic meter per second. It forms a pile
which has the shape of a cone. The height of the cone is equal to the radius of
the circle at its base. How fast is the sandpile rising when it is 2 meters high?

12. A water tank is in the shape of a cone with vertical axis and vertex downward.
The tank has radius 3 m and is 5 m high. At first the tank is full of water,
but at time t = 0 (in seconds), a small hole at the vertex is opened and the
water begins to drain. When the height of water in the tank has dropped to
3 m, the water is flowing out at 2 m3/s. At what rate, in meters per second,
is the water level dropping then?

7. A helicopter takes off from a point 80 m away from an observer located on the
ground, and rises vertically at 2 m/s. At what rate is elevation angle of the
observer’s line of sight to the helicopter changing when the helicopter is 60 m
above the ground.

8. An oil slick on a lake is surrounded by a floating circular containment boom.
As the boom is pulled in, the circular containment boom. As the boom is
pulled in, the circular containment area shrinks (all the while maintaining the
shape of a circle.) If the boom is pulled in at the rate of 5 m/min, at what
rate is the containment area shrinking when it has a diameter of 100m?

9. Consider a cube of variable size. (The edge length is increasing.) Assume that
the volume of the cube is increasing at the rate of 10 cm3/minute. How fast
is the surface area increasing when the edge length is 8 cm?

4. An airplane flying horizontally at an altitude of y = 3 km and at a speed of
480 km/h passes directly above an observer on the ground. How fast is the
distance D from the observer to the airplane increasing 30 seconds later?

5. A kite is rising vertically at a constant speed of 2 m/s from a location at
ground level which is 8 m away from the person handling the string of the
kite
(a) Let z be the distance from the kite to the person. Find the rate of change
of z with respect to time t when z = 10.
(b) Let x be the angle the string makes with the horizontal. Find the rate
of change of x with respect to time t when the kite is y = 6 m above
ground.

6. A balloon is rising at a constant speed 4m/sec. A boy is cycling along a
straight road at a speed of 8m/sec. When he passes under the balloon, it is
36 metres above him. How fast is the distance between the boy and balloon
increasing 3 seconds later.