Homework help?!

Question

1. If you were to grow plants in a rectangular-shaped garden with a perimeter of 10 ft, what is the maximum area you can achieve?

2. Assume that the function H(t) gives the height in ft of a stone thrown vertically upward at a velocity of 20 ft/s. How high does the stone go?

3. Back at work as a clown, you are pumping gas into a spherical balloon so that its volume increases at a rate of 2 in3/s. What is the rate of change of its radius when it is 6 inches?

4. Assume that ABC Airlines has a policy that states the baggage must be box-shaped and its sum of height, width and length must not exceed 108 inches. 1) What dimensions (length, width, height) would give the maximum volume and 2) what is the maximum volume?

Answer 1

QUESTION 1:

 

    Since the perimeter of the rectangle is 10, 2 of the consecutive sides is half that length, or 5 ft.

    Let "W" represent the width of the rectangle.

    Let "5 - W" represent the length of the triangle ... the sum of the 2 sides must be 5 ... W + 5 - W = 5

    The area of the rectangle is represented by the expression ... (W)(5 - W) ... or -W^2 + 5W

 

    The maximum area occurs at the vertex of this quadratic formula.  This parabola opens down (vertex is a maximum)

    "a" = -1, "b" = 5

    The x coordinate of the vertex occurs at --b/(2a) = -5/(2(-1)) = -5/(-2) = 2.5

    The y coordinate of the vertex is the value for the maximum value.  We compute that by plugging in 2.5 for "x".

    

    -(2.5)^2 + 5(2,5) = 6.25

 

    So the maximum area is 6,25 feet, and it occurs when one of the side lengths is 2,5 feet ... and since the sum of the 2 sides must be 5, both sides are 2.5 ft.