Differentiation - Lifting Rate Problem

Category: Differential Calculus, Plane Geometry

"Published in Newark, California, USA"

A weight is attached to one end of a 33-foot rope which passes over a pulley 18 feet above the ground. The other end is attached to a truck at a point 3 feet above the ground. If the truck moves away at a rate of 2 feet per second, how fast is the weight rising when the truck is 8 feet from the spot directly under the pulley?

Solution: 

To visualize the problem, let's draw the figure as follows

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As a truck moves away from the weight, a right triangle is formed between the weight, pulley, and a truck 

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By Pythagorean Theorem

In this problem, y is a constant because the height of a pulley from the ground is already fixed. As a truck moves away from the weight, the length of a rope from the truck to the pulley increases. Take the derivative of the above equation with respect to time t as follows

In order to get the value of ^dc/_dt, we need to solve for the value of c first because it is not given in the problem. Since the distance of a truck from the spot directly under the pulley, the height of a truck from the ground, and the height of a pulley from the ground are given, we can solve for the value of c which is the length of a rope from the truck to the pulley as follows

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By Pythagorean Theorem

The rate of increasing the length of a rope as a truck moves away from the spot directly under the pulley is also the same as the rate of rising the weight from the ground. Therefore, the rate of rising the weight from the ground is

                                        

Integration - Trigonometric Substitution

Category: Integral Calculus, Trigonometry, Algebra

"Published in Newark, California, USA"

Evaluate

Solution:

If you examine the given equation, we cannot integrate it by simple integration. Since the given equation has √ 4 - x2 , then we have to use the Trigonometric Substitution. Trigonometric Substitution is applicable if a function contains √ a2 - x2 , √ a2 + x2 , and √ x2 - a2 . 


First, draw a right triangle to represent √ 4 - x2  as follows

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Let

Substitute these values to the given equation, we have

Substitute the values of θ, Sin θ, and Cos θ to the above equation, we have

Substitute the value of the limits to the above equation, we have

Therefore,

Volume - Cube, Given Diagonal

Category: Solid Geometry, Plane Geometry

"Published in Newark, California, USA"

A diagonal of a cube joints two vertices not in the same face. If the diagonals are 4√3 cm. long, what is the volume?

Solution:

To visualize the problem, let's draw the figure as follows

Photo by Math Principles in Everyday Life

We know that all sides of a cube are equal because all faces of a cube are square. All sides of a cube are perpendicular to each other. A diagonal is a line segment that connects the two opposite vertices of a cube. There are 4 equal diagonals in a cube: AG, CE, BH, and FD.

How do you get the length of a diagonal of a cube if one side of a cube is given? Here's the procedure in getting the length of a diagonal of a cube as follows

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By Pythagorean Theorem

After we get the diagonal of a base, we can finally get the diagonal of a cube as follows

Photo by Math Principles in Everyday Life

By Pythagorean Theorem

The length of a diagonal of a cube is equal to the length of a side of a cube times square root of three. From the given word problem that if the length of a diagonal of a cube is 4√3 cm., then the length of a side of a cube will be 

Finally, we can get the volume of a cube as follows