Chapter 8.2, Problem 53PFA

a.

To determine

Tofind: The number of square units of fabric she needs to make each pennant.

Answer to Problem 53PFA

A. $x^3+1.5x^2+3x$

Explanation of Solution

Given:

Each pennant is of same size in the form of triangle with base $x$ and height $2x^2+3x+6$ .

Calculation:

To find the units of fabric required, find the area of the triangle with base $x$ and height $2x^2+3x+6$ .

Thus, the number of square units of fabric required is:

  $\begin{array}{c}\frac{1}{2}\cdot \left(\text{base}\right)\cdot \left(\text{height}\right)=\frac{1}{2}\cdot x\cdot \left(2x^2+3x+6\right)\\ =\frac{2x^3+3x^2+6x}{2}\\ =x^3+1.5x^2+3x\end{array}$

Therefore, the correct option is A. $x^3+1.5x^2+3x$ .

b.

To determine

To find: The number of square units of fabric required to make all pennants.

Answer to Problem 53PFA

C. $30x^3+45x^2+90x$

Explanation of Solution

Given:

Each pennant is of same size in the form of triangle with base $x$ and height $2x^2+3x+6$ .

There are 30 classmates.

Calculation:

The number of square units of fabric required to make each pennant is $x^3+1.5x^2+3x$ .

So, the number of square units of fabric required to make all the pennants is:

  $\begin{array}{c}30\left(x^3+1.5x^2+3x\right)\\ =30x^3+45x^2+90x\end{array}$

Therefore, the correct option is C. $30x^3+45x^2+90x$ .

c.

To determine

To find: The area one of the largest pennants if the base is 2 meters.

Answer to Problem 53PFA

  $20{\text{ m}}^2$

Explanation of Solution

Given:

The base of the pennant is 2 meters.

Calculation:

Substitute $x=2$ in the expression $x^3+1.5x^2+3x$ to find the area of the pennant.

  $\begin{array}{c}A={\left(2\right)}^3+1.5{\left(2\right)}^2+3\left(2\right)\\ =8+1.5\cdot 4+6\\ =8+6+6\\ =20{\text{ m}}^2\end{array}$

Therefore, the area of the one of the largest pennants is $20{\text{ m}}^2$ .

d.

To determine

To find: The total area of the fabric required to make 30 pennants for the city.

Answer to Problem 53PFA

  $600{\text{ m}}^2$

Explanation of Solution

Given:

The base of the pennant is 2 meters.

Calculation:

Substitute $x=2$ in the expression $30x^3+45x^2+90x$ to find the total area of the fabric required.

  $\begin{array}{c}\text{Total Area }=30{\left(2\right)}^3+45{\left(2\right)}^2+90\left(2\right)\\ =30\cdot 8+45\cdot 4+180\\ =240+180+180\\ =600{\text{ m}}^2\end{array}$

e.

To determine

To find: The total cost of the fabric she will need to make the pennants for the city if the fabric costs $\$10/{\text{m}}^2$ .

Answer to Problem 53PFA

  $\$6000$

Explanation of Solution

Given:

The cost of fabric is $\$10/{\text{m}}^2$ .

Calculation:

Total fabric required to make the pennants for the city is $600{\text{ m}}^2$ .

Therefore, the total cost of the fabric required to make pennants for the city is:

  $600{\text{ m}}^2\times \$10/{\text{m}}^2=\$6000$ .