Question

Write a function rule for the area A of a triangle whose base, b, is 2 cm less than seven times the height, h. What is the area of the triangle when the height is 14 cm?

Answer 1

`A(h)=(7)/(2)h^(2)-h(cm)`

The area of the triangle is
`672cm^(2)` when the height is 14 cm.

Explanation:

We start by finding the function rule for the area A.

We know reading the exercise that ''The base b is 2 cm less than seven times the height h''

Let's turn this into an equation :

`b=7h-2cm` (I)

The area of a triangle is
`(b.h)/(2)` (II)

Given that ''b'' is the base and ''h'' is the height.

Now we replace (I) in (II).

`A=(b.h)/(2)=((7h-2cm).h)/(2)=(7h^(2)-2h(cm))/(2)=(7)/(2)h^(2)-h(cm)`

The area A is a function of the variable ''h''

`A(h)=(7)/(2)h^(2)-h(cm)`

That is the function rule for the area A.

To find the area when the height is 14 cm we replace in the function rule h by 14 cm ⇒

`A(14cm)=(7)/(2)(14cm)^(2)-14cm(cm)=686cm^(2)-14cm^(2)=672cm^(2)`

The area of the triangle is
`672cm^(2)` when the height is 14 cm.

Answer 2

Let h be the height of the triangle. With this the base may be represented as 7h - 2. The area of the triangle is half the product of its base and height. Therefore,
A = (1/2)bh = (1/2)(7h - 2)h
Substitute to the equation the value of h,
A = (1/2)((7)(14 cm) - 2)(14 cm)
A = 672 cm²