Inclined Plane Problems

Category: Mechanics, Physics

"Published in Newark, California, USA"

The body on the inclined plane is subjected to the vertical and horizontal forces as shown in the figure. Find the components of each force along x-y axes oriented parallel and perpendicular to the incline.

Photo by Math Principles in Everyday Life

Solution:

Draw the x and y components of the given forces and then isolate the given figure as follows

Photo by Math Principles in Everyday Life

For F = 400 lbs.,

For P = 1200 lbs.,

Word Problem - Number Problem, 5

Category: Algebra

"Published in Daly City, California, USA"

The sum of the digits of a two-digit number is 7. When the digits are reversed, the number is increased by 27. Find the number.

Solution:

From the description of a given word problem, it is an application of linear equation with two equations, two unknowns. 

Let x = be the value of one's digit
      y = be the value of ten's digit 

From the statement "The sum of the digits of a two-digit number is 7.", then the working equation is
 

From the statement "When the digits are reversed, the number is increased by 27.", then the working equation is
 

 

 

 

Hence, the working equations for the given problem are
 

If you add the two working equation, y will be cancelled and the value of x which is the one's digit is
 

 

Substitute the value of x to any of the working equations. The value of y which is the ten's digit is
 

 

 

 

Therefore, the value of a two-digit number is 25. 

Check:

Word Problem - Number Problem, 4

Category: Algebra

"Published in Vacaville, California, USA"

The sum of two numbers is twice their difference. The larger number is 6 more than twice the smaller. Find the numbers.

Solution:

From the description of a given word problem, it is an application of linear equation with one variable.

Let x = be the value of a smaller number
      2x + 6 = be the value of a larger number

From the statement "The sum of two numbers is twice their difference.", then the working equation is

  
From the working equation, we can solve for the value of x which is the value of a smaller number as follows

The value of a larger number is

Therefore, the numbers are 6 and 18. 

Word Problem - Mixture Problem

Category: Algebra

"Published in Vacaville, California, USA"

A customer in a coffee shop wishes to purchase a blend of two coffees: Kenyan, costing $ 3.50 a pound, and Sri Lankan, costing $ 5.60 a pound. He ends up buying 3 lbs of such as blend, which costs him $11.55. How many pounds of each kind went into the mixture?

Solution:

From the description of a given word problem, it is an application of linear equation with two equations, two unknowns.  

Let x be the number of pounds of Kenyan coffee
      y be the number of pounds of Sri Lankan coffee

From the statement "He ends up buying 3 lbs of such as blend,..", then the working equation is

From the statement "A customer in a coffee shop wishes to purchase a blend of two coffees: Kenyan, costing $ 3.50 a pound, and Sri Lankan, costing $ 5.60 a pound....which costs him $11.55.", then the working equation is
 

Hence, the working equations for the given problem are
 

From the first equation,
 

 

Substitute the value of x to the second equation, we have
 

 

Substitute the value of y to any of two equations, we have

 

Therefore, a customer needs to buy a mixture of 2.5 pounds of Kenyan coffee and 0.5 pounds of Sri Lankan coffee that cost $11.55.