Runaway Car Using Hooke’s Law, we can show that thework done in compressing a spring a distance of x feetfrom its at-rest position is W = 12 kx2, where k is a stiffnessconstant depending on the spring. It can also be shown thatthe work done by a body in motion before it comes to restis given by W∼ = w2g v2, where w = weight of the object (lb),g = acceleration due to gravity (32.2 ft/sec2), andv = object’s velocity (in ft/sec). A parking garage has aspring shock absorber at the end of a ramp to stop runawaycars. The spring has a stiffness constant k = 9450 lb/ft andmust be able to stop a 4000-lb car traveling at 25 mph. Whatis the least compression required of the spring? Expressyour answer using feet to the nearest tenth.[Hint: Solve W 7 W∼, x Ú 0].
Optimization
Optimization comes from the same root as "optimal". "Optimal" means the highest. When you do the optimization process, that is when you are "making it best" to maximize everything and to achieve optimal results, a set of parameters is the base for the selection of the best element for a given system.
Integration
Integration means to sum the things. In mathematics, it is the branch of Calculus which is used to find the area under the curve. The operation subtraction is the inverse of addition, division is the inverse of multiplication. In the same way, integration and differentiation are inverse operators. Differential equations give a relation between a function and its derivative.
Application of Integration
In mathematics, the process of integration is used to compute complex area related problems. With the application of integration, solving area related problems, whether they are a curve, or a curve between lines, can be done easily.
Volume
In mathematics, we describe the term volume as a quantity that can express the total space that an object occupies at any point in time. Usually, volumes can only be calculated for 3-dimensional objects. By 3-dimensional or 3D objects, we mean objects that have length, breadth, and height (or depth).
Area
Area refers to the amount of space a figure encloses and the number of square units that cover a shape. It is two-dimensional and is measured in square units.
Runaway Car Using Hooke’s Law, we can show that the
work done in compressing a spring a distance of x feet
from its at-rest position is W = 1
2 kx2
, where k is a stiffness
constant depending on the spring. It can also be shown that
the work done by a body in motion before it comes to rest
is given by W
∼ = w
2g v2
, where w = weight of the object (lb),
g = acceleration due to gravity (32.2 ft/sec2
), and
v = object’s velocity (in ft/sec). A parking garage has a
spring shock absorber at the end of a ramp to stop runaway
cars. The spring has a stiffness constant k = 9450 lb/ft and
must be able to stop a 4000-lb car traveling at 25 mph. What
is the least compression required of the spring? Express
your answer using feet to the nearest tenth.
[Hint: Solve W 7 W
∼, x Ú 0].
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