Chapter 6, Problem 13GR

a.

To determine

To determine an expression for the height of the triangle.

Answer to Problem 13GR

  $2x-1$ centimeter

Explanation of Solution

Given:

Height of the triangle is $1$ centimeter less than the twice of the length of the base.

  $x=$Length of the base

Calculation:

Height of the triangle is $1$ centimeter less than the twice of the length of the base.

Considering the length of the base as $x$ .

So, height of the triangle is $2x-1$ centimeter.

Conclusion:

Therefore, expression for the height of the triangle is $2x-1$ centimeter.

b.

To determine

To determine a function rule for the area of the given triangle.

Answer to Problem 13GR

  $f(x)=x^2-\frac{x}{2}{\text{centimeter}}^2$

Explanation of Solution

Given:

Height of the triangle is $1$ centimeter less than the twice of the length of the base.

  $x=$Length of the base

Formula used:

Area of triangle $=\frac{1}{2}\times \text{base}\times \text{height}$

Calculation:

Height of the triangle is $1$ centimeter less than the twice of the length of the base.

  $x=$Length of the base So, height of the triangle is $2x-1$ centimeter.

Putting the values in above formula,

  $\Rightarrow$ Area of triangle $=\frac{1}{2}\times x\times (2x-1)$

  $\Rightarrow$ Area of triangle $=\frac{1}{2}\times (2x^2-x)$

  $\Rightarrow$ Area of triangle $=x^2-\frac{x}{2}$

Let’s consider area of triangle as $y$ .

  $y$ can be written as $f(x)$

  $\Rightarrow f(x)=x^2-\frac{x}{2}$

Conclusion:

Therefore, a function rule for the area of the triangle is $f(x)=x^2-\frac{x}{2}{\text{centimeter}}^2$ .

c.

To determine

To determine area of given triangle if the base is 16 centimeter.

Answer to Problem 13GR

  $248{\text{ centimeter}}^2$

Explanation of Solution

Given:

Base is 16 centimeter

Formula used:

Area of triangle $=\frac{1}{2}\times \text{base}\times \text{height}$

Calculation:

From the previous subpart function rule for the area of the given triangle is $f(x)=x^2-\frac{x}{2}{\text{centimeter}}^2$ .

Now, base is 16 centimeter, that is $x=16$ centimeter.

  $\Rightarrow$$f(16)={16}^2-\frac{16}{2}{\text{centimeter}}^2$

  $\Rightarrow$$f(16)=256-8{\text{centimeter}}^2$

  $\Rightarrow$$f(16)=248{\text{ centimeter}}^2$

Conclusion:

Therefore, area of given triangle if the base is 16centimeter is $248{\text{ centimeter}}^2.$