Solve the quadratic problem for me!

Question

Determine the maximum area of a triangle, in square centimeters, if the sum of its base and its height is 15 cm. 

Answer 1

A=(1/2)bh

b+h=15

b=15-h

substitute

A=(1/2)(15-h)(h)

A=(1/2)(15h-h²)

A=(15/2)(h)-(1/2)(h²)

the parabola opens downward so its vertex is the maximum point, (h,k), where h=-b/2a and k=area

-b/2a=(-15/2)/(2*[-1/2])

-b/2a=(-15/2)/(-1)

-b/2a=15/2=7.5

b+h=15

b+7.5=15

b=15-7.5=7.5

h=7.5 cm and b=7.5 cm

A=(1/2)(7.5)(7.5)

A=(1/2)(56.25)

A=28.125 sq cm is the maximum area

Answer 2

A triangle has exactly 3 sides.  Area is base times height divided by 2.

 

Let x = base

Let y = height

 

Set up an equation to represent the information.

 

x + y = 15                  eq1

(1/2)xy  = Area           eq2

 

 

Substitute eq1 into eq2.  We can get eq2 in terms of x and form a quadratic function.

 

Area = (1/2)x(15 - x)

 

Area = (1/2)(-x² + 15x)

 

Area = (-1/2)x² + 7.5x

 

 

Because the coefficient of the leading term is negative, this ensures that we will have a maximum.

 

To find the maximum of the area function we need to find the vertex (h, k).

 

h = -b / 2a

 

where:

a = -0.5

b = 7.5

 

 

Plug in these values into the formula for h.

 

h = -7.5 / (2*-0.5)

h = 7.5

 

This is the x-coordinate of the maximum.  Substitute this value into eq2  to get the maximum area.

 

Area = (-1/2)(7.5)² + (7.5)(7.5)

Area = 28.125

 

Maximum area is 28.125 cm².