Finding Last Coefficient - Quadratic Equation

Category: Algebra

"Published in Newark, California, USA"

Find the value of k to the given equation

if the product of the roots is -28.

Solution:

Consider the Quadratic Formula

If the roots of the quadratic equation are

Then

Substitute the value of the variables to the above equation

Therefore the answer is k = 28. 

Finding Middle Coefficient - Quadratic Equation

Category: Algebra

"Published in Newark, California, USA"

Find the value of k to the given equation 

if the sum of the roots is 2.

Solution:

Consider the Quadratic Formula

If the roots of the quadratic equation are

Then

Substitute the value of the variables to the above equation

Therefore, the answer is k = 2. 

Right Circular Cone Problem

Category: Solid Geometry

"Published in Newark, California, USA"

Two vertical conical tanks (both inverted) have their vertices connected by a short horizontal pipe. One tank, initially full of water, has an altitude of 6 feet and a diameter of base 7 feet. The other tank, initially empty, has an altitude of of 9 feet and a diameter of base 8 feet. If the water is allowed to flow through the connecting pipe, find the level to which the water will ultimately rise in the empty tank. (Neglect the water in the pipe.)

Solution:

Initial:

Photo by Math Principles in Everyday Life

Final:

Photo by Math Principles in Everyday Life

Consider the initial figure, 

 
Now, if the water is allowed to flow through the connecting pipe, then the final height of the first tank is equal to the height of the second tank after it is raised.

Consider the vertical section of the first tank at the final figure. We found out that the initial radius and height of a tank is similar to the final radius and height of a tank. Using similar triangles, we have

 
Let R₁ be the final radius. Therefore,

Consider the vertical section of the second tank at the final figure. We found out that the radius and height of an empty tank is similar to the final radius and height of a tank after the water is raised. Using similar triangles, we have

 
Let R₂ be the final radius. Therefore,

 
Equate equations 1 and 2, we have
 

 

 

                                            
Final Volume of the first tank,

Final Volume of the second tank,

 

Final Volume of the first tank + Final Volume of the second tank = Initial Volume of the first tank

we can cancel π as their common factor,

but 

Hence, the above equation becomes

Therefore, the final height of water is