quadratic equation?

Question

Write a quadratic equation for each of the following.

1. The area of an isosceles right triangle in terms of its legs.
2. The lateral area of a right circular cylinder whose height is equal to the radius.

thanks

Answer 1

Question 1
Let legs be x , x and h where h is the hypotenuse.
Area = (1/2) x²

Question 2
Circumference = 2 π r where r is the radius.
In this example, height h = r
Lateral area = 2 π r h = 2 π r ² units ²

Answer 2

#1) a isosceles right triangle's 2 equal sides -- a & b -- would be the ones that surround the right angle. Therefore, the area would be:

Area = a·b/2

and since they're equal, you can substitute one for the other.

Area = a·a/2 = a²/2 = (1/2)a²
or
Area = b·b/2 = b²/2 = (1/2)b²

Either of these is a quadratic equation, since there is a variable squared in each.

#2) The lateral area is the circumference of the base times the height:

LA = C·h

Circumference is diameter times π. Diameter is 2 times radius. Therefore, circumference is 2πr.
The height is equal to the radius, so h = r

Substituting:
LA = 2πr·r or 2πr²

The last one is the quadratic equation, since it has the term with a squared variable.

Answer 3

1. An isoceles right triangle is half of a square of sides equal to the triangle leg lengths. If the legs are each L, the area is simple (1/2)L^2.

2. [2(pi)(r)][r] = 2pir^2