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 cm^{2} 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).
__Solution to Problem 5:__
Using half the angle a, we can write
sin(a / 2) = (L / 2) / r
Substitute r by 10 and solve for L
L = 20 sin(a / 2)
The relationship between arc length s and central angle a is
s = r a = 10 a
Solve for a
a = s / 10
Substitute a by s / 10 in L = 20 sin(a / 2) to obtain
L = 20 sin ( (s / 10) / 2 )
= 20 sin ( s / 20)
__Problem 6:__ Express the distance d = d1+ d2, in the figure below, as a function of x.
__Solution to Problem 6:__
d1 is the length of the hypotenuse of a right triangle of sides x and 3, hence
d1 = sqrt[ 3^{ 2} + x^{ 2} ]
d2 is the length of the hypotenuse of a right triangle of sides 7 - x and 5, hence
d2 = sqrt[ 5^{ 2} + (7 - x)^{ 2} ]
d = d1 + d2 is given by
d = sqrt[ 9 + x^{ 2} ] + sqrt[ 25 + (7 - x)^{ 2} ]
__Exercises__
1. Express the area A of a disk in terms of its circumference C.
2. The width of a rectangle is w. Express the area A of this rectangle in terms of its perimeter P and width w.
__Solutions to above exercises__
1. A = C^{ 2} / (4 Pi)
2. A = (1/2) w (P - 2w)
More tutorials on functions.
Questions on Functions (with Solutions). Several questions on functions are presented and their detailed solutions discussed.
Applications, Graphs, Domain and Range of Functions |