|
Matched Problem 1: A rectangle has a perimeter of 60 m and an area of 200 m2. Find the length x and width y, x > y, of the rectangle.
Solution to Matched Problem 1:
- The perimeter of the rectangle is 60 m, hence
2x + 2y = 60
- The area of the rectangle is 200 m2, hence
x*y = 200
- Solve the equation 2x + 2y = 60 for y.
y = 30 - x
- Substitute y in the equation x*y = 200 by the expression for y obtained above.
x(30 - x) = 200
- Multiply, group like terms and write the above equation with the right hand side equal to zero.
-x2 +30x - 200 = 0
- Find the discriminant of the above quadratic equation.
Discriminant D = b2 - 4*a*c = 900 - 800 = 100
- Use the quadratic formulas to solve the quadratic equation; two solutions
x1 = [ -b + sqrt(D) ] / 2*a = [ -30 + 10 ] / -2 = 10 m
x2 = [ -b - sqrt(D) ] / 2*a = [ -30 - 10 ] / -2 = 20 m
- use y = 30 - x found above to find the corresponding value of y.
y1 = 30 - 10 = 20 m
y2 = 30 - 20 = 10 m
- Taking into account the condition x > y, the length x = 20 m and the width y = 10 m.
As an exercise, check the perimeter and the area.
Matched Problem 2: The sum of the squares of two consecutive even real numbers is 52. Find the numbers.
Solution to Problem 2:
- Let x and x+2 be the two consecutive even numbers. The sum of the square of x and x + 2 is equal to 52, hence
x2 + (x + 2)2 = 52
- Expand (x + 2)2, group like terms and write the above equation with the right side equal to zero.
2x2 + 4x - 48 = 0
- Multiply all terms in the above equation by 1/2.
x2 + 2x - 24 = 0
- Find the discriminant of the above quadratic equation.
Discriminant D = b2 - 4*a*c = 4 + 90 = 100
- Use the quadratic formulas to solve the quadratic equation; two solutions
x1 = [ -b + sqrt(D) ] / 2*a = [ -2 + 10 ] / 2 = 4
x2 = [ -b - sqrt(D) ] / 2*a = [ -2 - 10 ] / 2 = -6
- First solution to the problem
first number: x1 = 4
second number: x1 + 2 = 6
- Second solution to the problem
first number: x2 = -6
second number: x2 + 2 = -4
As an exercise check that the square of the two numbers, for each solution, is 52.
More references and links on how to Solve Equations, Systems of Equations and Inequalities.
|