A cable is to be run from a power plant on one side of a river 600 m wide to the factory on the other side 2 km downstream. The cost of running the cable over land is $8/m while the cost under water is $10/m. Find the total length of the cable at minimum cost.
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.
- A cable is to be run from a power plant on one side of a river 600 m wide to the factory on the other side 2 km downstream. The cost of running the cable over land is $8/m while the cost under water is $10/m. Find the total length of the cable at minimum cost.
- A stone is dropped in a pond. The circumference of the first ripple formed increases at a rate of 5o m/min. How fast is the area of the ripple increasing when the circumference of the circle is 100π m?
- An inverted conical tank is 200 cm in diameter at the base and 300 cm high. Water is drained from the tank at at the rate of 180π cm3/min. Determine the rate at which the top surface area of the water is shrinking when its radius is 60 cm.
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