Examples of applications of functions where quantities such area, perimeter, chord are expressed as function of a variable.
Problem 1: A right triangle has one side x and a hypotenuse of 10 meters. Find the area of the triangle as a function of x.
Solution to Problem 1:
If the sides of a right triangle are x and y, the area A of the triangle is given by
A = ( 1 / 2) x * y
We now need to express y in terms of x using the hypotenuse, side x and pythagora's theorem
10 2 = x 2 + y 2
y = sqrt [100 - x 2 ]
Substitute y by its expression in the area formula to obtain
A(x) = ( 1 / 2) x sqrt [100 - x 2 ]
Problem 2: A rectangle has an area equal to 100 cm2 and a width x. Find the perimeter as a function of x.
Solution to Problem 2:
If x and y are the dimensions of the rectangle, using the formula of the area we obtain
100 = x * y
The perimetr P is given by
P = 2(x + y)
Solve the equation 100 = x * y for y and substitute y in the formula for the perimeter
P(x) = 2(x + 100 / x)
Problem 3: Find the area of a square as a function of its perimeter x.
Solution to Problem 3:
The area of a square of side L is given by
A = L 2
The perimetr x of a square with side L is given by
x = 4 L
Solve the above for L and substitute in the area formula A above
A(x) = (x/4) 2 = x 2 / 16
Problem 4: A right circular cylinder has a radius r and a height equal to twice r. Find the volume of the cylinder as a function of r.
Solution to Problem 4:
The volume V of a right circular cylinder is given by
V = (area of base of cylinder) * (height of cylinder)
= Pi * r 2 * (2 r)
= 2 Pi r 3
Problem 5: Express the length L of the chord of a circle, with given radius r = 10 cm , as a function of the arc length s.(see figure below).