Chapter 8.6, Problem 85STP

To determine

To find: The height of the triangular sail.

Answer to Problem 85STP

The height of the triangular sail is $32x^3+8x^2-24x+16$ .

Explanation of Solution

Given information:

The area of triangular sail is $16x^4-60x^3-28x^2+56x-32$ .

The base of the triangle is $x-4$ meters.

Calculation:

Consider the triangular sail of area $f(x)$ square meters where:

  $f(x)=16x^4-60x^3-28x^2+56x-32$

Consider the base of the triangular sail is $(x-4)$ meters and height is $h$ meters.

Therefore,

  $f(x)=\frac{1}{2}(x-4)h$

Consider,

  $\begin{array}{c}f(x)=16x^4-64x^3-28x^2+56x-32\\ =16x^4+64x^3+4x^3-16x^2-12x^2+48x+8x-32\\ =16x^3(x-4)+4x^2(x-4)-12x^2(x-4)+8(x-4)\\ =(x-4)(16x^3+4x^2-12x+8)\end{array}$

Thus,

  $\begin{array}{c}\frac{1}{2}(x-4)h=(16x^3+4x^2-12x+8)\\ h=32x^3+8x^2-24x+16\end{array}$

Therefore, the height of the triangular sail is $32x^3+8x^2-24x+16$