problem solving

Question

1. The sum of two number is 10. The difference of their sqaures is 50. Calculate the TWN numbers. 2. Determine the length and the breadth of a rectangle if the length of a rectangle is 3m more than its breadth and the area if a rectangle is 50 meters squared. 3. The breadth of a rectangle is 3m less than its length. Calculate the length and the breadth of the rectangle if the area of the rectangle is 75 meters sqaured. 4. The hypotenuse of a right angled triangle is 10mm longer than the longest two other sides. Calculate the lengths of the sides if the shortest side is 50mm long.

Answer 1

1.

let a and b are the numbers.

The sum of two number is 10. --------> a + b = 10 --------> a = 10 - b

The difference of their sqaures is 5 ----------> a^2 - b^2 = 50

substitute a = 10 - b in a^2 - b^2 = 50

(10 - b)^2 - b^2 = 50

100 + b^2 - 20b - b^2 = 50

-20b = -50

b = 2.5

a = 10 - 2.5 = 7.5

Numbers are 7.5 and 2.5

Answer 2

2.

let l and b are the lenght and breadth of the rectangle.

length of a rectangle is 3m more than its breadth --------> l = 3 + w

area if a rectangle is 50 meters squared -------> lw = 50

substitute l = 3 + w in lw = 50

(3+w) * w = 50

3w + w^2 = 50

w = 5.72, -8.72

since width is measurement it can't be negative. w = 5.72 m

length = l = 3 + w = 3 + 5.72 = 8.72 m

Answer 3

3.

let l and b are the lenght and breadth of the rectangle.

length of a rectangle is 3m more than its breadth --------> w = l - 3

area if a rectangle is 50 meters squared -------> lw = 75

substitute l = 3 + w in lw = 75

(3+w) * w = 75

3w + w^2 =75

w = 7.289, -10.289

since width is measurement it can't be negative. w = 7.289 m

length = l = 3 + w = 3 + 7.289 = 10.889 m

Answer 4

4.

From Pythagorean trigonometric identity; hypotenuse^2 = side1^2 + side^2

h^2 = a^2  +b^2

The hypotenuse of a right angled triangle is 10mm longer than the longest two other sides -------> h = a + 10

lengths of the sides if the shortest side is 50mm long -------------> b = 50

(a + 10)^2 = a^2 + 50^2

a^2 + 100 + 20a = a^2 + 2500

20a = 2400

a = 120

hypotenuse = 120 + 10 = 130