this is a lot of questions. you're not trying to get us to do your homework are you?question is incomplete

hint: rate of change of distance with respect to time is given by the derivative. this gives the rate of change. you are asked to find t such that h'(t) = 75In the first seconds after launch, the height of a rocket above the ground is given by the expression:

h(t) = 5t^3 + 30t^2 + 45t + 4

where h is in metres and t is in seconds. How long does the rocket take to reach a speed of 75 m/s

NOTE: Speed is the rate of change of distance with respect to time

hint: start by drawing a diagram (ALWAYS a good idea for optimization problems, when possible, which it definitely is in this case). label the diagram. for instance, label the length x and the width (the side parallel to the existing fence) y. note thatA farmer has 140m of chicken wire which she is going to use against an existing boundary fence, to enclose a vegetable garden, which she wants in the shape of a rectangle.

Find the maximum area of her vegetable garden.

(You must justify, using calculus, that your result is the maximum area)

.............the total length of the wire. this is your constraint equation.

now, you want to maximize your area equation, given by:

hint: let r be the radius of the end of the can and h be the height of the can. as the problem directs,A drink manufacturer proposes a new style of can packaging. Given that the sum of the circumference and the height of the new can is to be 30cm, what is the maximum possible volume of the new can package?

(You must justify, using calculus, that your result is the maximum volume)

...........this is your constraint equation.

you want to maximize the volume, given by:

hint: differentiate the position equation twice to get the acceleration equation. you want the acceleration when t = 2A particle moves in a straight line so that its displacement from a point O, at any time t, is

x = *square root* 2t^3 + 4 (x is in metres and t is in seconds)

Find the acceleration when t = 2

(NOTE: Acceleration is the rate of change of velocity with respect to time)

[/quote]

The sum of two numbers is 10. What are the two numbers if the sum of their squares is a minimum?[/quote]let the numbers be x and y

the first sentence says,

x + y = 10 ..............this is your constraint equation

your objective equation is,

where S is the sum of the squares of the two numbers.

call y f(x) for clarity.For the curve, y = ax^2 + bx + c, where a, b and c are constants, it is given that at the points (2, 12) and (-1, 0) the slope of the tangent is 7 and 1 respectively.

Find a, b and c.

since (2,12) and (-1,0) are on the curve, we have:

f(2) = 12 and

f(-1) = 0

given the respective tangents, we have

f ' (2) = 7 and

f ' (-1) = 1

you can set up simultaneous equations with the information above to solve for a, b and c