Word Problem - Race Problem

Category: Algebra, Physics, Mechanics

"Published in Newark, California, USA"

In a 100-meter race, Pepito gives Willie a 30-meter start and losses by 2 ½  seconds. In the second trial, he gives Willie a 10-meter start and beats him by 2 ½ seconds. Find the rate of each runner.

Solution:

The given word problem above is about a marathon or race problem. Let's analyze the word problem as follows:

Let x = be the rate of Willie
      y = be the rate of Pepito
       t = total time of Pepito to finish the 100-meter race

If the first statement says "In a 100-meter race, Pepito gives Willie a 30-meter start and losses by 2 ½  seconds.", then the working equation will be

 
Note:

If the second statement says "In the second trial, he gives Willie a 10-meter start and beats him by 2 ½ seconds.", then the working equation will be

Equate the first equation with the second equation because the two cases or trials are both in a 100-meter race

Substitute the value of x to either of the two equations above, we have

Hence, the value of y will be equal to

Therefore, the final answers are

Rate of Pepito = y = 5 meters per seconds
Rate of Willie =  x = 4 meters per seconds
 

Algebraic Operations - Fractions

Category: Algebra

"Published in Newark, California, USA"

Perform the indicated operations for

Solution:

Consider the given equation above

Since the variables are all (x - y), then treat (x - y) as one variable. Factor the numerators and the denominators of the above equation as follows

Get the reciprocal of the divisor and perform the multiplication as follows

Cross out their common factor, we have

Therefore, the final answer is

Maximum Minimum Problem, 5

Category: Differential Calculus, Solid Geometry

"Published in Suisun City, California, USA"

A rectangular storage container with an open top is to have a volume of 10 m³. The length of its base is twice the width. Material for the base costs $10 per square meter. Material for the sides costs $6 per square meter. Find the cost of materials for the cheapest such container.

Solution:

To illustrate the problem, it is better to draw the figure as follows

Photo by Math Principles in Everyday Life

We know that the volume of a rectangular parallelepiped is

The area of the lower base is equal to

The area of the lateral sides is equal to

but

then the above equation becomes

The total cost of the material is equal to

Take the derivative on both sides of the equation with respect to x, we have

Set dC/dx = 0 because we want to minimize the cost of the material

Therefore, the total cost of the material is

Rationalize the denominator at the above equation, we have

Give the value of the cube roots of the above equation