6. The following problem was posed by Neal Koblitz in the March 1988 issue of the American Mathematical Monthly. The problem is one of several applied problems given to his calculus classes at the University of Washing- ton. Solve the problem using methods developed in this unit. ]A 75 feet HB You are standing on the ground at point B (see diagram), a distance of 75 ft from the bottom of a Ferris wheel with radius 20 ft. Your arm is at the same level as the bottom of the Ferris wheel. Your friend is on the Ferris wheel, which makes one revolution (counterclockwise) every 12 seconds. At the instant when she is at point A, you throw a ball to her at 60 ft/sec at an angle of 60° above the horizontal. Take g = -32 ft/sec?, and neglect air resistance. Find the closest distance the ball gets to your friend. (Source: American Mathematical Monthly, March 1988, page 256.)
Minimization
In mathematics, traditional optimization problems are typically expressed in terms of minimization. When we talk about minimizing or maximizing a function, we refer to the maximum and minimum possible values of that function. This can be expressed in terms of global or local range. The definition of minimization in the thesaurus is the process of reducing something to a small amount, value, or position. Minimization (noun) is an instance of belittling or disparagement.
Maxima and Minima
The extreme points of a function are the maximum and the minimum points of the function. A maximum is attained when the function takes the maximum value and a minimum is attained when the function takes the minimum value.
Derivatives
A derivative means a change. Geometrically it can be represented as a line with some steepness. Imagine climbing a mountain which is very steep and 500 meters high. Is it easier to climb? Definitely not! Suppose walking on the road for 500 meters. Which one would be easier? Walking on the road would be much easier than climbing a mountain.
Concavity
In calculus, concavity is a descriptor of mathematics that tells about the shape of the graph. It is the parameter that helps to estimate the maximum and minimum value of any of the functions and the concave nature using the graphical method. We use the first derivative test and second derivative test to understand the concave behavior of the function.
![6. The following problem was posed by Neal Koblitz in the March 1988 issue
of the American Mathematical Monthly. The problem is one of several
applied problems given to his calculus classes at the University of Washing-
ton. Solve the problem using methods developed in this unit.
]A
75 feet
HB
You are standing on the ground at point B (see diagram), a distance of 75 ft
from the bottom of a Ferris wheel with radius 20 ft. Your arm is at the same
level as the bottom of the Ferris wheel. Your friend is on the Ferris wheel,
which makes one revolution (counterclockwise) every 12 seconds. At the
instant when she is at point A, you throw a ball to her at 60 ft/sec at an angle
of 60° above the horizontal. Take g =-32 ft/sec?, and neglect air resistance.
Find the closest distance the ball gets to your friend. (Source: American
Mathematical Monthly, March 1988, page 256.)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fdaa299df-cb1e-4822-92c5-78198cbb1864%2Ff3c7936a-3e0f-4f73-b49a-1d1437be3237%2F1cbkeqf_processed.png&w=3840&q=75)
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