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Chapter 8: Applications of Derivatives
Solving Optimization Problems
498–512 Solve the given optimization problem.
Recall that a maximum or minimum value occurs
where the derivative is equal to zero, where the
derivative is undefined, or at an endpoint (if the
function is defined on a closed interval). Give an
exact answer, unless otherwise stated.
498. Find two numbers whose difference is 50
and whose product is a minimum.
499. Find two positive numbers whose product
is 400 and whose sum is a minimum.
500. Find the dimensions of a rectangle that has
a perimeter of 60 meters and whose area is
as large as possible.
501. Suppose a farmer with 1,500 feet of fencing
encloses a rectangular area and divides it
into four pens with fencing parallel to one
side. What is the largest possible total area
of the four pens?
502. A box with an open top is formed from a
square piece of cardboard that is 6 feet
wide. Find the largest volume of the box
that can be made from the cardboard.
494. A stone is thrown straight up with the
height given by the function s = 40t – 16t
2
,
where s is measured in feet and t is measured in seconds. What is the maximum
height of the stone? What is the velocity
of the stone when it’s 20 feet above the
ground on its way up? And what is its
velocity at that height on the way down?
Give exact answers.
495. A stone is thrown vertically upward with
the height given by s = 20t – 16t
2
, where s is
measured in feet and t is measured in seconds. What is the maximum height of the
stone? What is the velocity of the stone
when it hits the ground?
496. A particle moves on a vertical line so that
its coordinate at time t is given by y = t
3
– 4t
+ 5 for t ≥ 0. When is the particle moving
upward, and when is it moving downward?
Give an exact answer in interval notation.
497. A particle moves on a vertical line so that
its coordinate at time t is given by y =
4t
2
– 6t – 2 for t ≥ 0. When is the particle
moving upward, and when it is it moving
downward? Give your answer in interval
notation.
Chapter 8: Applications of Derivatives
Solving Optimization Problems
498–512 Solve the given optimization problem.
Recall that a maximum or minimum value occurs
where the derivative is equal to zero, where the
derivative is undefined, or at an endpoint (if the
function is defined on a closed interval). Give an
exact answer, unless otherwise stated.
498. Find two numbers whose difference is 50
and whose product is a minimum.
499. Find two positive numbers whose product
is 400 and whose sum is a minimum.
500. Find the dimensions of a rectangle that has
a perimeter of 60 meters and whose area is
as large as possible.
501. Suppose a farmer with 1,500 feet of fencing
encloses a rectangular area and divides it
into four pens with fencing parallel to one
side. What is the largest possible total area
of the four pens?
502. A box with an open top is formed from a
square piece of cardboard that is 6 feet
wide. Find the largest volume of the box
that can be made from the cardboard.
494. A stone is thrown straight up with the
height given by the function s = 40t – 16t
2
,
where s is measured in feet and t is measured in seconds. What is the maximum
height of the stone? What is the velocity
of the stone when it’s 20 feet above the
ground on its way up? And what is its
velocity at that height on the way down?
Give exact answers.
495. A stone is thrown vertically upward with
the height given by s = 20t – 16t
2
, where s is
measured in feet and t is measured in seconds. What is the maximum height of the
stone? What is the velocity of the stone
when it hits the ground?
496. A particle moves on a vertical line so that
its coordinate at time t is given by y = t
3
– 4t
+ 5 for t ≥ 0. When is the particle moving
upward, and when is it moving downward?
Give an exact answer in interval notation.
497. A particle moves on a vertical line so that
its coordinate at time t is given by y =
4t
2
– 6t – 2 for t ≥ 0. When is the particle
moving upward, and when it is it moving
downward? Give your answer in interval
notation.